Recent developments in finite rank systems
Donoso, Sebastián
I will comment on recent results concerning the topological properties of finite rank Cantor minimal systems. I will mention some ideas to estimate their word complexity and ask a few open problems.
 This talk is about some combinatorial, anthropological aspects of finite rank systems. I will start recalling the basic definitions and give some of the problematic. I will serve it a few results, then try to give some ideas on some special topics and then ask a few questions that we have. Please, if you have any questions, just let me know. Probably this will appear many times the basic definitions, so I will go more or less quickly. I will take a finite alphabet and I will here I will write the shift as the transformation S. I will not use the letter sigma because I will use that later. Capital S will be the shift, this is the shift map. Here I will be working with close subsets of a shift space that I will call generically, sorry here it should be S. I will call generically. Sorry. Hi, Sebastian. I think all I see are two arrows chasing themselves. We don't see the slides. Ah, you don't see the slides. Okay. Maybe only the organizers see. Okay, some people see. I can see. Okay, with my computer, sorry about that. Okay. Just let me know if there is any problem. Okay. So here we have I will use this generic notation to denote the sub-shift and as usual we say that the word, this is finite word, finite sequence of symbols appears in the shift if you can see that word in some point of your shift space. And this is also probably well known. The language of words of size length, N, is it's all the finite words that appear somewhere in X. And here this is one that would be one of the main objects we will study. This is the complexity, the complexity function. I apologize for my bad handwriting. This is the complexity function that counts the number of finite length words, the words of length N in the subject. And just to recall the topological entropy, I will write it like that, of the sub-shift is just the exponential growth rate of this function. This is the limit of the logarithm of this function and you divide that by N. This is the topological entropy. And this quantity, the complexity function, has been studied by several authors and some restricted cases, this complexity function tells us some things about the system itself. So let me go to the next one. And some motivation about this complexity function is that there are some complexity restrictions for systems regarding this complexity function. So probably the most well known result is when your system is finite, if not only if this complexity function is bounded. The next step is what I will call the non-superlinear complexity case, which will be when the lemmins, which will be when the lemmins of the complexity function divided by N, this is a finite number. So this will be the non-superlinear complexity. And this case poses several restrictions. And let me just start with the obvious one. The system, having this condition, they have zero entropy. This is just because the entropy is the exponential growth rate of this. And if this is, say, it has this growth rate condition, this cannot be positive. So this is almost immediate consequence. And less trivial consequence is that systems like this have a finite number of ergodic measures. So let's say finite number of ergodic measures. And here let me mention some names. So this was first observed by Bočer Niedsen in, I think, 1985, 84. We showed that this number here bounds, or this number, if this number is finite, then you have a finite number of ergodic measures. And here ergodic measures are all the invariant, so all the extreme points of the set of invariant measures. And this property has been studied more in details very recently. So let me mention there are some recent results by Van Sier and Braynacra. This is, I think, 2019, where they studied the number of ergodic measures and genetic measures and give a more precise bounds. I will just comment that this has been studied recently. Just to give a complete list of references, there is also a result by Damron and Fikentcher. I'm sorry if I'm pronouncing it wrongly. This is 2017. And I think this will be something related to their result will be in a talk tomorrow. So I advertise this talk about related results. And there is also a recent paper by Dijkstra, Ampavlov, where they extend results to transitive substitutes. So here everything is minimal. They study for, give more results for transitive systems. Okay, so this is just to say that systems with this condition have some rigidity properties. And I mean rigidity in a very vague sense. Rigidity is what I'm writing here. That they have zero entropy, they have a finite number of ergodic measures. And there is another property that this system has. And is that, I will maybe call it here. And that they have a restricted automorphism group or constrained automorphism group. Constraint automorphism group. And here the automorphism group of this shift is all the automorphisms, they call it H, all the automorphisms such that they commute with the shift action. So this is the automorphism group. And in these systems, this is constrained. And this is, there are also very recent results about this question. And I think to be more or less complete with the references, the first result in this direction was given by Salo and Torma. I think this is 2013, where they show that, maybe I will write here the condition. Is that if you consider the automorphism group and you quotient by the group generated by the shift, this is fine. So this is, it was first observed by, proved by Salo and Torma in 2013 for some classes of substitutions. And then we have Van Siet and Brian Acra. And also myself with Fabian, Fabian Durand, Alejandro Mas and Samuel Petit, where we show that non-superlinear complexity, so the condition having the limit, the complexity finite. This is the non-superlinear condition. This implies that this object is finite. The group of the automorphism divided by the shift, where the group generated by the shift is a finite group. And so that is, sorry, I forgot to mention here that is also a was, sorry, I forget if I write correctly and coven I think. Let's say around the same time. And actually all this result used a condition, I will call three prime, which is that this algebraic condition is a consequence of having finite number of asymptotic component, finite number of asymptotic components. And what is this? So two points in the sub-shift are asymptotic. If they are equal, they have equal values starting from some point. So the typical, this picture is some slay that. So starting from some point, the sequences coincide and after that point, they differ somewhere. Probably we can think that they differ there. And after that they are equal. And being asymptotic is an equivalence relation, so you can talk about asymptotic components. And the property behind this property three is that such systems with this condition have a finite number of asymptotic components. So to summarize this slide, we have this complexity restriction and we want to understand what rigidity properties implies. So here I mention three or four of them. Any questions so far? Sorry, is it working? Yes, okay. When you say finite number of asymptotic component, so do you mean like any configuration is not asymptotic to itself? Because if this holds then... Ah yes, yes, sorry. Non-trivial asymptotic. Non-trivial. Okay, so here I have some restrictions. Let me go up zero entropy, finite number of erythrofagotic measures constrained automorphism group and this finite number of asymptotic components, not trivial asymptotic components. So then we can ask if all these restrictions that appear here are really... maybe they are more general. Maybe they appear in some... with some topological assumption more general than this combinatorial property. So if you look at the restrictions one and two that I have here, zero entropy and finite number of erythrofagotic measures, one could guess a class of systems that one could try to understand to see if all... if those systems are actually... if that class of system is actually the one having all these restrictions that I mentioned here. So I have that and this is what we call the finite rank systems and to define finite rank systems which is the the title, I will have to introduce briefly what is called the Bratelli-Versic diagram. Probably there will be more talks about this, so I will spend a few minutes talking about this representation and then mention some results. Okay, so as I mentioned before, the motivation was to understand all these restrictions that the non-superlinear complexity gives and try to see if there is some more general class maybe defined not in that combinatorial terms as having non-superlinear complexity that also has all those rigidity properties and that's what will be the class of finite rank systems that I will introduce in a few more. Okay, so to do that I have to tell you about the Bratelli-Versic diagrams. So what is a Bratelli-Versic diagram? It's a graph, infinite graph where you can put different vertices in different levels and you can connect say a level with the next one and with nothing else. So you can you put together all vertices in a given level and then you have an infinite graph like this and the assumption is that here you have one special vertex which is the root of the diagram. So here I have vertices edges between levels and that's basically what is a Bratelli-Versic diagram. Okay, so between two levels I will call those levels and in principle I should name them all differently. So this should be 1 sub n minus 1 but to lighten the notation I will just write 1, 2, 3 etc. So these are what are called levels. You have in one level a set of vertices and in the next level you have another set of vertices, different vertices and what you have is you have some edges connecting them and that's the structure of a given level and you have that for infinitely many levels and this gives you a Bratelli-Dig. Okay and the next thing concerning these structures is for any given vertex you have an order in the edges that arrive to that vertex like the edges like that are pointing up say and here I wrote this order 1, 2, 3, 4 but this I will be considering this type of order but it's just for convenience. I will order them from left to the right but this could be any order could be 1, could be 1, 3, 2 etc. It can be any order but for the sake of the character I will just I will mainly use from left to the right and having done that you can create what's called the Bratelli-Versic map which is as follows and I will not describe it formally, I will just give a few examples and let's hope things will be clear. So given this Bratelli diagram you can associate the set of all infinite paths is the set of all infinite paths. When I mean paths I will read the edges of this graph this is an infinite path and we can define the Bratelli-Versic map as the successor map so I will give this example here and let's see if it's clear. This is actually the most easy example one can imagine. We will have one vertex at each level and we will connect with two edges to the next one so here this continues I just wrote a portion of it and here I'm drawing one path this one in green so the successor map will be I will look down till I find something that is not maximum here I'm also considering the order from left to right so this is smaller than this one so remember in each vertex I had an order that allows me to compare the edges that are arriving to this vertex so if we look at this path in green so the rule is I will look down till I find something that is not maximum so here if I start looking down I will find this edge that is not maximum so I will mark it it's not maximum and what I will do I will move it to the next one in the order I had so here I'm given an order I will move it to the next one so this will move to the next one and I erase everything I had above but I erase what I had before and to draw what I had here I will come back with the minimum path so I will choose at any step the minimum path that allows me to come back so in this case it's just that so this is the this is the successor map and if we think of this as the sequence of zeros and ones so where say this is a zero and this is one so the one in the left is a zero and the one in the right is one what we have done here is that let me go back is that here the the sequence will be one zero zero so what we did is that one zero zero something it went to sorry no I didn't have I have that one one zero zero it's okay one zero zero it went to zero one zero so this is an this can be seen as adding one in the in this daily group and what is adding one I add the coordinate and if and I keep the the how it's called the I forgot the name in English so carry here when I add one the carry yes sorry the I add the I add one here so this one becomes zero and I keep the carry here and I put it there and I continue like that so someone that if you haven't looked at this kind of dynamics if you have never looked at this then maybe you are asking yourself what happens if I never find the something that is not maximum well in that case you send to the path that is everything is a minimum of course if that is well defined if there is only one maximum and one okay but this is just to give a taste of what is a dynamics in this kind of maps this example here is what is called an odometer it's an odometer or adding machine so you have a group operation and what you're doing the successor dynamics is adding some fixed element so this is an odometer and what is the particular particular aspect of the odometer is that in each level you have one one vertex and actually that's what characterize the odometers in this context okay so and then I will just draw a few pictures that so you can imagine what happens when you have more complicated structure here the structure was quite simple just one vertex at each level but you can have much more complicated structures and the dynamics is exactly the same it's just more difficult to look at it but the idea is the same so here for instance let's consider this this path in green of course this continues down but I just draw a portion of it so the dynamics is you go down till you find something that you can move something that is not the maximum so in this case so this is maximum this is maximum and this is not maximum so I should be able to move that one so I mark that as non-the maximum and then I move it there and then I come back to the to the top following the minimum path so that's one iteration of the successor map and if your if your graph is complicated enough you will have a complicated enough dynamics so that's what I wanted to say about these maps any question not please let me know if you have any question okay so why one could one could ask after all these definitions what is the class of systems that can be built like that like that and here two important theorems are the following ones just to mention the condition that we will be using the diagram is simple if you can connect no to vertices that are far away in the diagram and this is the condition to have a minimal dynamic minimal dynamics means that every orbit is a dance is dancing in the space so say a fundamental result in this context is a theorem in the 92 by Herman Putman and Scho that show that actually any minimal counter dynamics any minimal counter system can be represented in this way so and here minimal counter system means that is a minimal topological system and this space is a counter space so this is a counter space so this is kind of a universality or represent representation theorem of all dynamics that happen in a counter system in a counter space all of them can be represented as I said before so the class of systems we can treat with these diagrams is quite big it's all counter systems and the definition we will be discussing next is the one of finite rank the one here finite rank so a system is a finite rank if I can represent it as previously with the diagram where the number of vertices doesn't increase to infinity so let me go back here here I could have that when I go more down the number of vertices can go to infinity and this is a valid system the systems of all the dynamics but in the case where this doesn't happen or when you can represent it in a way that this doesn't happen this is called a finite rank system so all systems that can be represented in this way and this class of system starts to exhibit some rigidity properties and the first one is this theorem by Tomas Nonarovich and Alejandro Mas from 2008 that says that if you have a system like this where you can or that can be represented with a diagram finite where the number of vertices is bounded then it's either expansive meaning a subshift or an odometer like the situation I draw in the beginning you have one vertex and you just you are just adding a fixed number the fixed element in that group so this is a dichotomy theorem that says everything in that class is a subshift or it's an a key continue system it's a rotation in on a compact delin group so here we have a first restriction to this class because if you are wondering counter space it can be all subshifts are happening in a counter space but you can have things that are not expansive so not everything is a subject okay so this is a very first result and this the theorem says that well more or less we can focus on on subshifts because the other case would be odometer and odometer is easier to understand and this class of systems was studied in details in 2012 by Besugli, Tiatowski, Medinet and Solomniac and they exhibited they exhibited some rigidity properties that I will briefly mention here and the first property they prove is that they have a finite number of ergodic measures finite number of ergodic measures as much as like in the case of non-superlina complexity actually the number of ergodic measures is bounded by the rank bounded less than the rank so the maybe I didn't mention the rank is like the number of vertices that you can use at each level to represent the system the minimum number and so the system have finite number of ergodic measures and they have zero entropy what they showed is that they have zero entropy for any invariant measure which of course implies that the vibrational principle that the topological entropy is zero so we have these rigidity conditions in this class which are the ones I mentioned in the beginning that happen for non-superlinear systems so here we can ask what how about the next the next restriction so let me and before presenting the next restriction I will put in context this class of systems with the the condition we had in non-superlina complexity so this is a recent part of a recent joint work with Fabian and Juran Alejandro Marz and Samurta Tit and we show that the condition of non-superlina complexity actually fits in this class meaning that if a system is like that has non-superlina complexity then it has to be a finite rank so ease of the form I described before so this gives an indirect they prove that this system have a finite number of ergodic measures for instance using the previous result I mentioned here in the this slide but it's very indirect it's not really optimal in the in the in the sense of the number so then one can ask very quickly if maybe this complexity condition characterizes this topological condition of finite rank and this is false actually even with rank two systems so this is a system that is given by two two vertices at each level in this class in a system like this you don't have you may not have a condition with the complexity so there are examples where the complexity grows more than linear so super linear complexity but is of a finite rank so just to draw a picture here we have the finite rank systems and here inside we have non-super linear this is non-super linear and this is uh these two classes are different and I mentioned in the beginning that this non-super linear class presents some restrictions like entropy number of ergodic measures which are also that also happens in the finite rank but they also have this automorphism group restrictions and asymptotic per restriction so the first question that one could ask is are those uh conditions also satisfied in finite rank system so let me show you some questions these are some questions that we post in the the paper but they are really solved so first question is does the minimal uh finite rank system have many many many finite many asymptotic components and this was solved by bastian spinosa which is a graduate student of alehandra and fabian and this this condition of sorry sorry there is a question of borisolomiaq he's asking how fast can the complexity grow for finite rank oh yeah very good question um I actually I I I wrote it down in the end so maybe we can talk about that again I I actually wrote it that's an open question I want to ask so this is um this was already solved this is I think 2020 and this of course implies that the automorphism group of these systems is also constrained and some other topological consequences or questions that is every factor of finite rank also finite rank this was not clear at all but this was also solved by uh uh let me go lestani bastiano also and go lestani I'm sorry if I pronounce it correctly and joseini this is also 2020 and um this was recently like last month solved by also by bastian spinosa that in terms of symbolic factors this class is also very restricted has finitely many of them have it has finitely many factors up to topological conjugacy and I advertise that we have here talks on on thursday so I advertise this talks that will be on thursday and I I think they will talk more in details about these results so I will just mention them I have another question and related to this all these topological questions one of the aspects I would like to treat in this class of systems is the complexity one so that that is what we are going to discuss more in details in what follows what is the complexity what is the complexity of finite rank systems we know it's a zero entropy but uh what is a possible uh complexities you can see and how how can you control them if I give you an example how how can you estimate the complexity and to do so I will introduce a few notions of morphisms and esadic subsets which are are a bit more convenient to to discuss this question so and here I will give a few more definitions I morphism will be any map I will call it say tow from an alphabet to another one that it's you can think of this as the concatenation of images of any symbols so tow of a sequence finite sequence say of a word it's just the concatenation of the the image of each letter that that's what I will call a morphism I will not call it a substitution because the alphabets may be different so to restrict the word substitution for same alphabet I will call this a morphism and given a sequence of morphisms here positive I will restrict to positive morphisms and positive means that if I write a letter say if I write tow of a I will see some letters b1 b2 etc I want to see all of them so all letters appear in the images of of the letters in the in the alphabet in the in the beginning this is the same as when you construct the matrix associated to this morphism you have a positive matrix and this is not a restrictive assumption all the systems we are looking at can be can be considered with this condition so it's not it's not anything it's not restrictive just to make things easier so we have a sequence of positive morphisms where the alphabets might be different and having that sequence of morphism we can build what is called the esadic subshift associated to the sequence of morphisms which is given by the following condition a point is in the subshift if the finite words of this point can be obtained iterating some letter from far from far away so let me try to do the picture I have a sequence of morphisms this is okay close from an even alphabet the next one and this goes to say tau zero here I have my original alphabet a0 alphabet a1 you have some way to send letters here into words here and the point in this subshift belongs to the this alphabet the first alphabet to the z and the point is in there if when I take any subword of it here I will take a subword this subword comes from iterating some letter in some say alphabet say that this is this is the ak and and you iterate this with all the morphisms and when you do that you have a word that sees this finding word other way to say it is that you start iterating morphisms from far and far away and the limit points you get are the points in the sxh so these are like a generalization of the substitution but you allow the alphabets and the morphisms to change every time so this is a way to to to build what is called the sxhift and what actually what it's proved in the recent results I mentioned is that this this is intimately related to the finite grand systems and how to relate this class of system with the finite grand systems is that finite grand systems are exactly the systems where you can control the size of the alphabet here the alphabet can grow can grow to infinity the size the size can grow to infinity but if you are if you restrict to morphisms that where the alphabet doesn't grow to infinity then you actually get the same the same class of finite grand system and this this will be explaining details I think in the talks on third because this is the it's a theorem it's not it's not it's not trivial I mean probably it's one can expect some some this type of result but there is some technical things to do in the middle okay so the next question is if I give if I give you the sequence of morphisms how can you compute the complexity or is there any way you can compute the complexity so this is the next I will spend the remaining minutes of my talk talking about this so how is how to estimate this complexity without when I give you an example or in some classes etc and well of course there are some non-case substitutions substitution is one that all the morphisms are equal to the same to given morphisms so there is only one morphism that you repeat infinitely many times and this case in this case the the the complexity sublinear meaning it's bounded it's a p of n it's bounded by constant times n another class that is well studied in in symbolic dynamics is the linearly recurrent subshift I will not give that like the original definition but this will be when the morphisms you have a finite set of finite set of morphisms so finite set of morphisms and in this case the complexity is also sublinear so it's a finite set of morphisms that you that you choose how you start iterating and then there is the finite rank which you can ask what are the complexities that are here but this is always our explanation because well it has zero entropy okay so the what we tried to develop was some ideas to compute the complexity in some examples or in some classes and I will mention you some of the tools we saw or we tried to create to to compute the complexity here everything I'm I'm talking about is minimal and the morphisms are positive okay and to understand how is the complexity we will have to look at these quantities so for a for a morphism let's denote the double bar of the morphism the the the maximal length of of tau of a letter so you write down so you have your letter a b c say you write the substitution and you may have that some letters give you a very long word some other letters give you a very short word so I will write with this the longest the longest length and with this symbol the shortest and and actually we'll see that these two quantities are behind the complexity so if you have a morphism where these two things are very different you can achieve very non-trivial complexities like high complexity okay so here there is the first tool to understand the complexity that is and is what we call the repetition complexity and okay the definition I will just write how to compute it suppose that you have a that gives you a a b c a b so what this what this complexity for this quantity means is that you will count for each letter you will count how many times you change the letter but here we started with b with a so I will start with one because I'm using a letter then here I change the letter I will plus one here I change another letter I will plus one here I change another well this is a bad example it doesn't illustrate what's happening maybe let's say that I have many b's and in dn and a so here I am not changing the letter here I change the letter I plus plus one so the this when I do this I count one two three four five six and I do that for every letter so b will give me something say seven and I add all those quantities and this is what we call the repetition complexity with how many times you can you need to change the letter in the in the morphism and that's should be related to the complexity because if you if you change too much you are creating many words somehow and that's exactly what it happens this is a computation that I will just write the like form of it but it can be very it can be done much more precisely and proposition with this quantity is the following suppose you have a sub shift then you can bound the complexity in an n or for a large enough n by so this is a it's not a precise bound but it will tell it will work to illustrate the result so you bound by the maximum of the alphabets of course this makes sense only if the I mean this is meaningful only in the case where the alphabets remain bounded and then you multiply this by the the link soup of this complexity that I defined previously so you start computing how you the morphism changed the letters and you have to multiply that by the size of the the maximum of the alphabets and this is the constant term that goes with n of course this can this can grow but this is meaningful only when this is bounded because it will give us some information when the system have sublinear complexity so if you have a static sub shift where we can bound these two things then the system has sublinear complexity sublinear complexity these are are quite strong conditions but I included it here just to illustrate how to use that notion but it can be it can be described much more precisely we can talk afterwards if you have a need and as a consequence of this result we did use some results in the equivalence strongly or with equivalence theory that says that if you have any finite topological rank then it's a strongly or with equivalent to something of sublinear complexity meaning I will just say it with words that means that you can send orbits to orbits in a somehow continuous way but this notion reserves for instance the set of infinite measures and some other under dimension group of course and this proposition gives for instance an alternative proof that the finite rank systems have finitely many agroic measures because you it's equivalent to something of sublinear complexity but of course there we lose the precise bound which is just an observation and so this is this complexity notion allows us to compute some things but actually there is more precise notion that we described that is the complexity of relative complexity of morphisms this is somehow how are how the morphisms in the aesthetic representation are being related so I think I have very few minutes so probably I will skip that part if you have if you want to take a look at this we can talk afterwards but this is a more precise notion to compute the complexity the previous one is easy to it's easy to compute in examples but it is highly non-optimal to compute the complexity this one is a bit better computing the complexity but it might be more complicated in examples but okay I will skip this because I'm running out of time okay and the consequence of these computations is that one can say when a system has zero entropy and it's it has to do with the the length the how the alphabet the exponential growth of the alphabets compared to the smallest word of the sequence of morphisms this is probably for poor result but we gave a very concise proof of this in particular finite rank have zero entropy okay so I wanted some other consequence that sub quadratic complexity is always true when you have two alphabets of size two and okay since I ran out of time let me go to the questions which I probably are more interesting so here are a few questions that are still open maybe the specialists can say if they make sense like people that have worked more on finite rank system so the first thing is if you can control in a polynomial way the complexity of a finite rank system so can it be can you find a degree that controls the polynomial the complexity of the finite rank system or maybe not and give an example where you cannot do that this is the first question so what I said previously that when the alphabet is of size two you can do it and the second question is if you can characterize complexity and having topological finite rank in some subclass of systems for instance we don't know if these conditions are equivalent in the topics class maybe in the topics class having finite topological rank is the same as having non-superlinear complexity there is some evidence that supports that but I will not mention that we can discuss it later if you have any questions and then we have a variety of other questions okay thank you Sebastian maybe we can pass to the questions there is a question in the chat room of Seijat Babel he's asking what can be saved the minimality condition is replaced by unique minimal subset in the in particular in the result of finite rank when the condition of non-superlinear complexity implies finite rankness if you have I'm not sure I I should replace I'm not sure like what happens if you drop the minimality probably you can say things but I haven't thought about it okay there is another question of Martin Lustig he's asking the chat and say what about for placing for subchips the water diversity rank by the by a nestadic rank defined through the existence of a nestadic development which is everywhere growing of buddha the level of a betranks and perhaps also recognized recognizable on each level this yeah um and you mean not not positive maybe I think actually this question of Martin will be it's one of the things that will be discussed on Thursday I think that's one is a recent paper of for Bastia and Spinoza where he addresses that question uh if I'm not mistaken I think there is some development of that other question a question of uh Dan Rust uh will you what's that the lens of each morphism is bounded what's happened in this case related to Martin uh Martin questions no they don't need to be bounded bounded they can the size can grow which will always happen if you telescope the diagram like if you look at it in the in the Bratelli version but you can you are allowed to do that okay so let's thank Sebastian
 This talk is about some combinatorial, anthropological aspects of finite rank systems.  I will start recalling the basic definitions and give some of the problematic. I will serve  it a few results, then try to give some ideas on some special topics and then ask a few questions  that we have. Please, if you have any questions, just let me know. Probably this will appear many  times the basic definitions, so I will go more or less quickly. I will take a finite alphabet and  I will here I will write the shift as the transformation S. I will not use the letter  sigma because I will use that later. Capital S will be the shift, this is the shift map.  Here I will be working with close subsets of a shift space that I will call generically,  sorry here it should be S. I will call generically. Sorry. Hi, Sebastian. I think all I see are two  arrows chasing themselves. We don't see the slides. Ah, you don't see the slides. Okay.  Maybe only the organizers see. Okay, some people see. I can see.  Okay, with my computer, sorry about that. Okay. Just let me know if there is any problem.  Okay. So here we have I will use this generic notation to denote the sub-shift and as usual  we say that the word, this is finite word, finite sequence of symbols appears in the shift if you  can see that word in some point of your shift space. And this is also probably well known. The  language of words of size length, N, is it's all the finite words that appear somewhere in X.  And here this is one that would be one of the main objects we will study. This is the complexity,  the complexity function. I apologize for my bad handwriting. This is the complexity function  that counts the number of finite length words, the words of length N in the subject. And just to  recall the topological entropy, I will write it like that, of the sub-shift is just the exponential  growth rate of this function. This is the limit of the logarithm of this function and you divide  that by N. This is the topological entropy. And this quantity, the complexity function,  has been studied by several authors and some restricted cases, this complexity function  tells us some things about the system itself. So let me go to the next one.  And some motivation about this complexity function is that there are some complexity  restrictions for systems regarding this complexity function. So probably the most  well known result is when your system is finite, if not only if this complexity function is bounded.  The next step is what I will call the non-superlinear complexity case, which will be  when the lemmins, which will be when the lemmins of the complexity function divided by N,  this is a finite number. So this will be the non-superlinear complexity. And this case  poses several restrictions. And let me just start with the obvious one. The system, having this  condition, they have zero entropy. This is just because the entropy is the exponential  growth rate of this. And if this is, say, it has this growth rate condition, this cannot be positive.  So this is almost immediate consequence. And less trivial consequence is that systems like this have  a finite number of ergodic measures. So let's say finite number of ergodic measures.  And here let me mention some names. So this was first observed by Bočer Niedsen  in, I think, 1985, 84. We showed that this number here bounds, or this number, if this number is  finite, then you have a finite number of ergodic measures. And here ergodic measures are all the  invariant, so all the extreme points of the set of invariant measures. And this property has been  studied more in details very recently. So let me mention there are some recent results by  Van Sier and Braynacra. This is, I think, 2019, where they studied the number of ergodic measures  and genetic measures and give a more precise bounds. I will just comment that this has been  studied recently. Just to give a complete list of references, there is also a result by Damron and  Fikentcher. I'm sorry if I'm pronouncing it wrongly. This is 2017. And I think this will be  something related to their result will be in a talk tomorrow. So I advertise this talk  about related results. And there is also a recent paper by Dijkstra,  Ampavlov, where they extend results to transitive substitutes. So here everything is  minimal. They study for, give more results for transitive systems. Okay, so this is just to  say that systems with this condition have some rigidity properties. And I mean rigidity in a very  vague sense. Rigidity is what I'm writing here. That they have zero entropy, they have a finite  number of ergodic measures. And there is another property that this system has. And is that,  I will maybe call it here. And that they have a restricted automorphism group or constrained  automorphism group. Constraint automorphism group. And here the automorphism group of  this shift is all the automorphisms, they call it H, all the automorphisms such that they  commute with the shift action. So this is the automorphism group. And in these systems,  this is constrained. And this is, there are also very recent results about this question.  And I think to be more or less complete with the references, the first result in this direction  was given by Salo and Torma. I think this is 2013, where they show that,  maybe I will write here the condition. Is that if you consider the automorphism group  and you quotient by the group generated by the shift, this is fine.  So this is, it was first observed by, proved by Salo and Torma in 2013 for some classes of  substitutions. And then we have Van Siet and Brian Acra. And also myself with Fabian,  Fabian Durand, Alejandro Mas and Samuel Petit, where we show that non-superlinear complexity,  so the condition having the limit, the complexity finite. This is the non-superlinear condition.  This implies that this object is finite. The group of the automorphism divided by the  shift, where the group generated by the shift is a finite group.  And so that is, sorry, I forgot to mention here that is also a  was, sorry, I forget if I write correctly and coven I think.  Let's say around the same time. And actually all this result used a condition,  I will call three prime, which is that this algebraic condition is a consequence of having  finite number of asymptotic component, finite number of asymptotic  components. And what is this? So two points in the sub-shift are asymptotic. If they are equal,  they have equal values starting from some point. So the typical, this picture is some  slay that. So starting from some point, the sequences coincide and after that point,  they differ somewhere. Probably we can think that they differ there. And after that they are equal.  And being asymptotic is an equivalence relation, so you can talk about asymptotic components.  And the property behind this property three is that such systems with this condition  have a finite number of asymptotic components. So to summarize this slide, we have this  complexity restriction and we want to understand what rigidity properties implies. So here I mention  three or four of them. Any questions so far?  Sorry, is it working? Yes, okay. When you say finite number of asymptotic component,  so do you mean like any configuration is not asymptotic to itself? Because if this holds then...  Ah yes, yes, sorry. Non-trivial asymptotic.  Non-trivial.  Okay, so here I have some restrictions. Let me go up zero entropy, finite number of  erythrofagotic measures constrained automorphism group and this finite number of asymptotic  components, not trivial asymptotic components. So then we can ask if all these restrictions  that appear here are really... maybe they are more general. Maybe they appear in some...  with some topological assumption more general than this combinatorial property.  So if you look at the restrictions one and two that I have here, zero entropy and finite  number of erythrofagotic measures, one could guess a class of systems that one could try to  understand to see if all... if those systems are actually... if that class of system is actually the  one having all these restrictions that I mentioned here. So I have that and this is  what we call the finite rank systems and to define finite rank systems which is the  the title, I will have to introduce briefly what is called the Bratelli-Versic diagram.  Probably there will be more talks about this, so I will spend a few minutes talking about  this representation and then mention some results. Okay, so as I mentioned  before, the motivation was to understand all these restrictions that the non-superlinear  complexity gives and try to see if there is some more general class maybe defined not in  that combinatorial terms as having non-superlinear complexity that also has all those rigidity  properties and that's what will be the class of finite rank systems that I will introduce in a few  more. Okay, so to do that I have to tell you about the Bratelli-Versic diagrams. So what is a  Bratelli-Versic diagram? It's a graph, infinite graph where you can put different vertices in  different levels and you can connect say a level with the next one and with nothing else. So you  can you put together all vertices in a given level and then you have an infinite graph like this  and the assumption is that here you have one special vertex  which is the root of the diagram. So here I have vertices edges between levels and that's  basically what is a Bratelli-Versic diagram. Okay, so between two levels I will call those  levels and in principle I should name them all differently. So this should be 1 sub n minus 1  but to lighten the notation I will just write 1, 2, 3 etc. So these are what are called levels.  You have in one level a set of vertices and in the next level you have another set of vertices,  different vertices and what you have is you have some edges connecting them  and that's the structure of a given level and you have that for infinitely many levels and  this gives you a Bratelli-Dig. Okay and the next thing concerning these structures is for any given  vertex you have an order in the edges that arrive to that vertex like the edges like that are pointing  up say and here I wrote this order 1, 2, 3, 4 but this I will be considering this type of order  but it's just for convenience. I will order them from left to the right but this could be any  order could be 1, could be 1, 3, 2 etc. It can be any order but for the sake of the character I will  just I will mainly use from left to the right and having done that you can create  what's called the Bratelli-Versic map which is as follows and I will not describe it formally,  I will just give a few examples and let's hope things will be clear.  So given this Bratelli diagram you can associate the set of all infinite paths  is the set of all infinite paths.  When I mean paths I will read the edges of this graph this is an infinite path  and we can define the Bratelli-Versic map as the successor map so I will give this example here  and let's see if it's clear. This is actually the most easy example one can imagine.  We will have one vertex at each level and we will connect with two edges to the next one so here  this continues I just wrote a portion of it and here I'm drawing one path this one in green  so the successor map will be I will look down till I find something that is not maximum  here I'm also considering the order from left to right so this is smaller than this one so  remember in each vertex I had an order that allows me to compare the edges that are  arriving to this vertex so if we look at this path in green  so the rule is I will look down till I find something that is not maximum  so here if I start looking down I will find this edge that is not maximum so I will mark it  it's not maximum and what I will do I will move it to the next one in the order I had  so here I'm given an order I will move it to the next one so this will move to the next one  and I erase everything I had above but I erase what I had before and to draw what I had here  I will come back with the minimum path so I will choose at any step the minimum path that allows me  to come back so in this case it's just that so this is the this is the successor map and if we  think of this as the sequence of zeros and ones so where say this is a zero and this is one so the  one in the left is a zero and the one in the right is one what we have done here is that let me go  back is that here the the sequence will be one zero zero so what we did is that one zero zero  something it went to sorry no I didn't have I have that one one zero zero it's okay one zero zero  it went to zero one zero so this is an this can be seen as adding one in the in this daily group  and what is adding one I add the coordinate and if and I keep the the how it's called the  I forgot the name in English so carry here when I add one the carry yes sorry  the I add the I add one here so this one becomes zero and I keep the carry here and I put it there  and I continue like that so someone that if you haven't looked at this kind of dynamics  if you have never looked at this then maybe you are asking yourself what happens if I never  find the something that is not maximum well in that case you send to the path that is  everything is a minimum of course if that is well defined if there is only one maximum and one  okay but this is just to give a taste of what is a dynamics in this kind of maps this example  here is what is called an odometer it's an odometer or adding machine so you have a group operation  and what you're doing the successor dynamics is adding some fixed element so this is an odometer  and what is the particular particular aspect of the odometer is that in each level you have one  one vertex and actually that's what characterize the odometers in this context  okay so and then I will just draw a few pictures that so you can imagine what happens when you  have more complicated structure here the structure was quite simple just one vertex at each level  but you can have much more complicated structures and the dynamics is exactly the same it's just  more difficult to look at it but the idea is the same so here for instance let's consider this  this path in green of course this continues down but I just draw a portion of it so the dynamics is  you go down till you find something that you can move something that is not the maximum  so in this case so this is maximum this is maximum and this is not maximum so I should be able to  move that one so I mark that as non-the maximum and then I move it there and then I come back  to the to the top following the minimum path so that's one iteration of the successor map  and if your if your graph is complicated enough you will have a complicated enough dynamics  so that's what I wanted to say about these maps any question  not please let me know if you have any question okay so why one could one could ask after all  these definitions what is the class of systems that can be built like that like that and  here two important theorems are the following ones just to mention the condition  that we will be using the diagram is simple if you can connect  no to vertices that are far away in the diagram and this is the condition to have a minimal  dynamic minimal dynamics means that every orbit is a dance is dancing in the space  so say a fundamental result in this context is a theorem in the 92 by Herman Putman and  Scho that show that actually any minimal counter dynamics any minimal counter system  can be represented in this way so and here minimal counter system means that is a minimal  topological system and this space is a counter space so this is a counter space  so this is kind of a universality or represent representation theorem of all dynamics that  happen in a counter system in a counter space all of them can be represented as I said before  so the class of systems we can treat with these diagrams is quite big it's all counter systems  and the definition we will be discussing next is the one of finite rank the one here finite rank  so a system is a finite rank if I can represent it as previously with the diagram where the number  of vertices doesn't increase to infinity so let me go back here here I could have that  when I go more down the number of vertices can go to infinity and this is a valid system  the systems of all the dynamics but in the case where this doesn't happen or when you can represent  it in a way that this doesn't happen this is called a finite rank system so all systems that can be  represented in this way and this class of system starts to exhibit some rigidity properties and  the first one is this theorem by Tomas Nonarovich and Alejandro Mas from 2008 that says that if you  have a system like this where you can or that can be represented with a diagram finite where the  number of vertices is bounded then it's either expansive meaning a subshift or an odometer  like the situation I draw in the beginning you have one vertex and you just you are just adding  a fixed number the fixed element in that group so this is a dichotomy theorem that says everything  in that class is a subshift or it's an a key continue system it's a rotation in on a compact  delin group so here we have a first restriction to this class because if you are wondering counter  space it can be all subshifts are happening in a counter space but you can have things that are not  expansive so not everything is a subject okay so this is a very first result and this  the theorem says that well more or less we can focus on on subshifts because the other case  would be odometer and odometer is easier to understand and this class of systems was studied  in details in 2012 by Besugli, Tiatowski, Medinet and Solomniac and they exhibited  they exhibited some rigidity properties that I will briefly mention here and the first  property they prove is that they have a finite number of ergodic measures  finite number of ergodic measures as much as like in the case of non-superlina complexity  actually the number of ergodic measures is bounded by the rank bounded less than the rank  so the maybe I didn't mention the rank is like the number of vertices that you can use at each level  to represent the system the minimum number and so the system have finite number of ergodic measures  and they have zero entropy what they showed is that they have zero entropy for any invariant  measure which of course implies that the vibrational principle that the topological entropy is zero  so we have these rigidity conditions in this class  which are the ones I mentioned in the beginning that happen for non-superlinear systems  so here we can ask what how about the next the next restriction  so let me and before presenting the next restriction I will put in context  this class of systems with the the condition we had in non-superlina complexity  so this is a recent part of a recent joint work with Fabian and Juran Alejandro Marz  and Samurta Tit and we show that the condition of non-superlina complexity actually fits in this  class meaning that if a system is like that has non-superlina complexity  then it has to be a finite rank so ease of the form I described before  so this gives an indirect they prove that this system have a finite number of ergodic measures  for instance using the previous result I mentioned here in the this slide  but it's very indirect it's not really optimal in the in the in the sense of the number  so then one can ask very quickly if maybe this complexity condition characterizes  this topological condition of finite rank and this is false  actually even with rank two systems so this is a system that is given by two  two vertices at each level in this class in a system like this you don't have  you may not have a condition with the complexity so there are examples where the complexity grows  more than linear so super linear complexity but is of a finite rank  so just to draw a picture here we have the finite rank systems  and here inside we have non-super linear  this is non-super linear  and this is uh these two classes are different and I mentioned in the beginning that this  non-super linear class presents some restrictions like entropy number of ergodic measures which are also  that also happens in the finite rank but they also have this  automorphism group restrictions and asymptotic per restriction so the first question that  one could ask is are those uh conditions also satisfied in finite rank system so let me show you  some questions these are some questions that we post in the the paper but they are really solved  so first question is does the minimal uh finite rank system have many many  many finite many asymptotic components and this was solved by bastian spinosa  which is a graduate student of alehandra and fabian and this this condition of sorry sorry  there is a question of borisolomiaq he's asking how fast can the complexity grow for finite rank  oh yeah very good question um I actually I I I wrote it down in the end so maybe we can  talk about that again I I actually wrote it that's an open question I want to ask  so this is um this was already solved this is I think 2020 and this of course implies that the  automorphism group of these systems is also constrained  and some other topological consequences or questions that is every factor of finite rank  also finite rank this was not clear at all but this was also solved by uh uh let me go lestani bastiano  also and go lestani I'm sorry if I pronounce it correctly and joseini  this is also 2020 and um this was recently like last month solved by also by bastian spinosa  that in terms of symbolic factors this class is also very restricted has finitely many of them  have it has finitely many factors up to topological conjugacy and I advertise that  we have here talks on on thursday so I advertise  this talks that will be on thursday and I I think they will talk more in details about  these results so I will just mention them I have another question  and related to this all these topological questions one of the aspects I would like to treat in this  class of systems is the complexity one so that that is what we are going to discuss  more in details in what follows what is the complexity  what is the complexity of finite rank systems  we know it's a zero entropy  but uh what is a possible uh complexities you can see and how how can you control them if I give  you an example how how can you estimate the complexity and to do so I will introduce a few  notions of morphisms and esadic subsets which are are a bit more convenient to to discuss this question  so and here I will give a few more definitions I  morphism will be any map I will call it say tow from an alphabet to another one that it's  you can think of this as the concatenation of images of any symbols so  tow of a sequence finite sequence say  of a word it's just the concatenation of the  the image of each letter that that's what I will call a morphism I will not call it a  substitution because the alphabets may be different so to restrict the word substitution for  same alphabet I will call this a morphism and given a sequence of morphisms  here positive I will restrict to positive morphisms and positive means that if I write a letter  say if I write tow of a I will see some letters b1 b2 etc I want to see all of them  so all letters appear in the images of of the letters in the in the alphabet in the  in the beginning this is the same as when you construct the matrix associated to this morphism  you have a positive matrix and this is not a restrictive assumption  all the systems we are looking at can be can be considered with this condition so it's not  it's not anything it's not restrictive just to make things easier so we have a sequence of  positive morphisms where the alphabets might be different and having that sequence of morphism  we can build what is called the esadic subshift associated to the sequence of morphisms  which is given by the following condition a point is in the subshift if  the finite words of this point can be obtained iterating some letter from far  from far away so let me try to do the picture I have a sequence of morphisms  this is okay close from an even alphabet  the next one and this goes to say tau zero here I have my original alphabet a0  alphabet a1 you have some way to send letters here into words here  and the point in this subshift belongs to the this alphabet the first alphabet to the z and  the point is in there if when I take any subword of it here I will take a subword  this subword comes from iterating  some letter in some say alphabet say that this is this is the ak and  and you iterate this with all the morphisms  and when you do that you have a word that sees this finding word  other way to say it is that you start iterating morphisms from far and far away  and the limit points you get are the points in the sxh  so these are like a generalization of the substitution but you allow the alphabets  and the morphisms to change every time so this is a way to to to build what is called the sxhift  and what actually what it's proved in the recent results I mentioned  is that this this is intimately related to the finite grand systems and how to relate  this class of system with the finite grand systems is that finite grand systems are exactly  the systems where you can control the size of the alphabet here the alphabet can grow  can grow to infinity the size the size can grow to infinity but if you are if you restrict to  morphisms that where the alphabet doesn't grow to infinity then you actually get the same  the same class of finite grand system and this this will be explaining details I think in the talks  on third because this is the it's a theorem it's not it's not it's not trivial I mean probably it's  one can expect some some this type of result but there is some technical things to do in the middle  okay so the next question is if I give if I give you the sequence of morphisms  how can you compute the complexity or is there any way you can compute the complexity  so this is the next I will spend the remaining minutes of my talk talking about this  so how is how to estimate this complexity  without when I give you an example or in some classes etc and well of course there are some  non-case substitutions substitution is one that all the morphisms are equal to the same to given  morphisms so there is only one morphism that you repeat infinitely many times and this case in  this case the the the complexity sublinear meaning it's bounded it's a p of n  it's bounded by constant times n  another class that is well studied in  in symbolic dynamics is the linearly recurrent subshift I will not give that like the original  definition but this will be when the morphisms you have a finite set of finite set of morphisms  so finite set  of morphisms  and in this case the complexity is also sublinear  so it's a finite set of morphisms that you that you choose how you start iterating  and then there is the finite rank which you can ask what are the complexities that are here but  this is always our explanation because well it has zero entropy  okay so the what we tried to  develop was some ideas to compute the complexity in some examples or in some classes  and I will mention you some of the tools we saw or we tried to create  to to compute the complexity  here everything I'm I'm talking about is minimal and the morphisms are positive  okay and to understand how is the complexity we will have to look at these quantities  so for a for a morphism let's denote the double bar of the morphism the  the the maximal length of of tau of a letter so you write down so you have your letter a b c say  you write the substitution  and you may have that some letters give you a very long word some other letters give you a very  short word so I will write with this the longest the longest length and with this symbol the shortest  and and actually we'll see that these two quantities  are behind the complexity so if you have a morphism where these two things are very different  you can achieve very non-trivial complexities like high complexity okay so here there is the first  tool to understand the complexity that is and is what we call the repetition complexity and  okay the definition I will just write how to compute it suppose that you have a  that gives you a a b c a b so what this what this  complexity for this quantity means is that you will count for each letter you will count  how many times you change the letter but here we started with b with a so I will start with one  because I'm using a letter then here I change the letter I will plus one here I change another  letter I will plus one here I change another well this is a bad example it doesn't illustrate  what's happening maybe let's say that I have many b's and in dn and a  so here I am not changing the letter here I change the letter I plus plus one so the  this when I do this I count one two three four five six and I do that for every letter  so b will give me something say seven and I add all those quantities and this is what we  call the repetition complexity with how many times  you can you need to change the letter in the in the morphism and that's should be related  to the complexity because if you if you change too much you are creating many words somehow  and that's exactly what it happens this is a computation  that I will just write the like form of it but it can be very it can be done much more precisely  and proposition with this quantity is the following suppose you have a sub shift  then you can bound the complexity in an n or for a large enough n by so this is a it's not  a precise bound but it will tell it will work to illustrate the result so you bound by the  maximum of the alphabets of course this makes sense only if the I mean this is meaningful  only in the case where the alphabets remain bounded and then you multiply this by the the  link soup of this complexity that I defined previously so you start computing  how you the morphism changed the letters and you have to multiply that by the size of the  the maximum of the alphabets and this is the constant term that goes with n of course this  can this can grow but this is meaningful only when this is bounded because it will give us  some information when the system have sublinear complexity so if you have a static sub shift  where we can bound these two things then the system has sublinear complexity sublinear complexity  these are are quite strong conditions but I included it here just to illustrate  how to use that notion but it can be it can be described much more precisely  we can talk afterwards if you have a need and as a consequence of this result we did use some  results in the equivalence strongly or with equivalence theory that says that  if you have any finite topological rank then it's a strongly or with equivalent to something of  sublinear complexity meaning I will just say it with words that means that you can send orbits  to orbits in a somehow continuous way but this notion reserves for instance the set of infinite  measures and some other under dimension group of course and this proposition gives for instance  an alternative proof that the finite rank systems have finitely many agroic measures  because you it's equivalent to something of sublinear complexity but of course there we lose  the precise bound which is just an observation and  so this is this complexity notion allows us to compute some things but actually there is  more precise notion that we described that is the complexity of relative complexity of morphisms  this is somehow how are how the morphisms in the aesthetic representation are being related  so I think I have very few minutes so probably I will skip that part if you have if you want to  take a look at this we can talk afterwards but this is a more precise notion to compute the  complexity the previous one is easy to it's easy to compute in examples but it is highly  non-optimal to compute the complexity this one is a bit better computing the complexity but  it might be more complicated in examples but okay I will skip this because I'm running out of time  okay and the consequence of these computations is that one can say when a system has zero entropy  and it's it has to do with the the length the how the alphabet the exponential growth of the  alphabets compared to the smallest word of the sequence of morphisms this is probably for  poor result but we gave a very concise proof of this in particular finite rank have zero entropy  okay so I wanted some other consequence that sub quadratic complexity is always true when you have  two alphabets of size two  and okay since I ran out of time let me go to the questions which I probably are more interesting  so here are a few questions that are still open maybe the specialists can say if they make sense  like people that have worked more on finite rank system so the first thing is if you can  control in a polynomial way the complexity of a finite rank system  so can it be can you find a degree that controls the polynomial the complexity of the finite  rank system or maybe not and give an example where you cannot do that this is the first question  so what I said previously that when the alphabet is of size two you can do it  and the second question is if you can characterize complexity  and having topological finite rank in some subclass of systems for instance  we don't know if these conditions are equivalent in the topics class maybe in the topics class  having finite topological rank is the same as having non-superlinear complexity  there is some evidence that supports that but I will not mention that we can discuss it later  if you have any questions and then we have a variety of other questions okay thank you  Sebastian maybe we can pass to the questions  there is a question in the chat room of Seijat Babel  he's asking what can be saved the minimality condition is replaced by unique minimal subset  in the in particular in the result of finite rank when the condition of non-superlinear  complexity implies finite rankness  if you have  I'm not sure I  I  should replace  I'm not sure like what happens if you drop the minimality probably you can say things but I  haven't thought about it okay there is another question of Martin Lustig  he's asking the chat  and say what about for placing for subchips the water diversity rank by the by a nestadic  rank defined through the existence of a nestadic development which is everywhere growing of  buddha the level of a betranks and perhaps also recognized recognizable on each level  this yeah  um and you mean not not positive maybe I think actually this question of Martin will be it's  one of the things that will be discussed on Thursday I think that's one is a recent paper of  for Bastia and Spinoza where he addresses that question  uh if I'm not mistaken I think there is some development of that  other question  a question of uh Dan Rust uh will you  what's that the lens of each morphism is bounded what's happened in this case  related to Martin uh Martin questions  no they don't need to be bounded bounded  they can the size can grow which will always happen if you telescope the diagram  like if you look at it in the in the Bratelli version  but you can you are allowed to do that  okay so let's thank Sebastian
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This talk is about some combinatorial, anthropological aspects of finite rank systems. I will start recalling the basic definitions and give some of the problematic. I will serve it a few results, then try to give some ideas on some special topics and then ask a few questions that we have. Please, if you have any questions, just let me know. Probably this will appear many times the basic definitions, so I will go more or less quickly. I will take a finite alphabet and I will here I will write the shift as the transformation S. I will not use the letter sigma because I will use that later. Capital S will be the shift, this is the shift map. Here I will be working with close subsets of a shift space that I will call generically, sorry here it should be S. I will call generically. Sorry. Hi, Sebastian. I think all I see are two arrows chasing themselves. We don't see the slides. Ah, you don't see the slides. Okay. Maybe only the organizers see. Okay, some people see. I can see. Okay, with my computer, sorry about that. Okay. Just let me know if there is any problem. Okay. So here we have I will use this generic notation to denote the sub-shift and as usual we say that the word, this is finite word, finite sequence of symbols appears in the shift if you can see that word in some point of your shift space. And this is also probably well known. The language of words of size length, N, is it's all the finite words that appear somewhere in X. And here this is one that would be one of the main objects we will study. This is the complexity, the complexity function. I apologize for my bad handwriting. This is the complexity function that counts the number of finite length words, the words of length N in the subject. And just to recall the topological entropy, I will write it like that, of the sub-shift is just the exponential growth rate of this function. This is the limit of the logarithm of this function and you divide that by N. This is the topological entropy. And this quantity, the complexity function, has been studied by several authors and some restricted cases, this complexity function tells us some things about the system itself. So let me go to the next one. And some motivation about this complexity function is that there are some complexity restrictions for systems regarding this complexity function. So probably the most well known result is when your system is finite, if not only if this complexity function is bounded. The next step is what I will call the non-superlinear complexity case, which will be when the lemmins, which will be when the lemmins of the complexity function divided by N, this is a finite number. So this will be the non-superlinear complexity. And this case poses several restrictions. And let me just start with the obvious one. The system, having this condition, they have zero entropy. This is just because the entropy is the exponential growth rate of this. And if this is, say, it has this growth rate condition, this cannot be positive. So this is almost immediate consequence. And less trivial consequence is that systems like this have a finite number of ergodic measures. So let's say finite number of ergodic measures. And here let me mention some names. So this was first observed by Bočer Niedsen in, I think, 1985, 84. We showed that this number here bounds, or this number, if this number is finite, then you have a finite number of ergodic measures. And here ergodic measures are all the invariant, so all the extreme points of the set of invariant measures. And this property has been studied more in details very recently. So let me mention there are some recent results by Van Sier and Braynacra. This is, I think, 2019, where they studied the number of ergodic measures and genetic measures and give a more precise bounds. I will just comment that this has been studied recently. Just to give a complete list of references, there is also a result by Damron and Fikentcher. I'm sorry if I'm pronouncing it wrongly. This is 2017. And I think this will be something related to their result will be in a talk tomorrow. So I advertise this talk about related results. And there is also a recent paper by Dijkstra, Ampavlov, where they extend results to transitive substitutes. So here everything is minimal. They study for, give more results for transitive systems. Okay, so this is just to say that systems with this condition have some rigidity properties. And I mean rigidity in a very vague sense. Rigidity is what I'm writing here. That they have zero entropy, they have a finite number of ergodic measures. And there is another property that this system has. And is that, I will maybe call it here. And that they have a restricted automorphism group or constrained automorphism group. Constraint automorphism group. And here the automorphism group of this shift is all the automorphisms, they call it H, all the automorphisms such that they commute with the shift action. So this is the automorphism group. And in these systems, this is constrained. And this is, there are also very recent results about this question. And I think to be more or less complete with the references, the first result in this direction was given by Salo and Torma. I think this is 2013, where they show that, maybe I will write here the condition. Is that if you consider the automorphism group and you quotient by the group generated by the shift, this is fine. So this is, it was first observed by, proved by Salo and Torma in 2013 for some classes of substitutions. And then we have Van Siet and Brian Acra. And also myself with Fabian, Fabian Durand, Alejandro Mas and Samuel Petit, where we show that non-superlinear complexity, so the condition having the limit, the complexity finite. This is the non-superlinear condition. This implies that this object is finite. The group of the automorphism divided by the shift, where the group generated by the shift is a finite group. And so that is, sorry, I forgot to mention here that is also a was, sorry, I forget if I write correctly and coven I think. Let's say around the same time. And actually all this result used a condition, I will call three prime, which is that this algebraic condition is a consequence of having finite number of asymptotic component, finite number of asymptotic components. And what is this? So two points in the sub-shift are asymptotic. If they are equal, they have equal values starting from some point. So the typical, this picture is some slay that. So starting from some point, the sequences coincide and after that point, they differ somewhere. Probably we can think that they differ there. And after that they are equal. And being asymptotic is an equivalence relation, so you can talk about asymptotic components. And the property behind this property three is that such systems with this condition have a finite number of asymptotic components. So to summarize this slide, we have this complexity restriction and we want to understand what rigidity properties implies. So here I mention three or four of them. Any questions so far? Sorry, is it working? Yes, okay. When you say finite number of asymptotic component, so do you mean like any configuration is not asymptotic to itself? Because if this holds then... Ah yes, yes, sorry. Non-trivial asymptotic. Non-trivial. Okay, so here I have some restrictions. Let me go up zero entropy, finite number of erythrofagotic measures constrained automorphism group and this finite number of asymptotic components, not trivial asymptotic components. So then we can ask if all these restrictions that appear here are really... maybe they are more general. Maybe they appear in some... with some topological assumption more general than this combinatorial property. So if you look at the restrictions one and two that I have here, zero entropy and finite number of erythrofagotic measures, one could guess a class of systems that one could try to understand to see if all... if those systems are actually... if that class of system is actually the one having all these restrictions that I mentioned here. So I have that and this is what we call the finite rank systems and to define finite rank systems which is the the title, I will have to introduce briefly what is called the Bratelli-Versic diagram. Probably there will be more talks about this, so I will spend a few minutes talking about this representation and then mention some results. Okay, so as I mentioned before, the motivation was to understand all these restrictions that the non-superlinear complexity gives and try to see if there is some more general class maybe defined not in that combinatorial terms as having non-superlinear complexity that also has all those rigidity properties and that's what will be the class of finite rank systems that I will introduce in a few more. Okay, so to do that I have to tell you about the Bratelli-Versic diagrams. So what is a Bratelli-Versic diagram? It's a graph, infinite graph where you can put different vertices in different levels and you can connect say a level with the next one and with nothing else. So you can you put together all vertices in a given level and then you have an infinite graph like this and the assumption is that here you have one special vertex which is the root of the diagram. So here I have vertices edges between levels and that's basically what is a Bratelli-Versic diagram. Okay, so between two levels I will call those levels and in principle I should name them all differently. So this should be 1 sub n minus 1 but to lighten the notation I will just write 1, 2, 3 etc. So these are what are called levels. You have in one level a set of vertices and in the next level you have another set of vertices, different vertices and what you have is you have some edges connecting them and that's the structure of a given level and you have that for infinitely many levels and this gives you a Bratelli-Dig. Okay and the next thing concerning these structures is for any given vertex you have an order in the edges that arrive to that vertex like the edges like that are pointing up say and here I wrote this order 1, 2, 3, 4 but this I will be considering this type of order but it's just for convenience. I will order them from left to the right but this could be any order could be 1, could be 1, 3, 2 etc. It can be any order but for the sake of the character I will just I will mainly use from left to the right and having done that you can create what's called the Bratelli-Versic map which is as follows and I will not describe it formally, I will just give a few examples and let's hope things will be clear. So given this Bratelli diagram you can associate the set of all infinite paths is the set of all infinite paths. When I mean paths I will read the edges of this graph this is an infinite path and we can define the Bratelli-Versic map as the successor map so I will give this example here and let's see if it's clear. This is actually the most easy example one can imagine. We will have one vertex at each level and we will connect with two edges to the next one so here this continues I just wrote a portion of it and here I'm drawing one path this one in green so the successor map will be I will look down till I find something that is not maximum here I'm also considering the order from left to right so this is smaller than this one so remember in each vertex I had an order that allows me to compare the edges that are arriving to this vertex so if we look at this path in green so the rule is I will look down till I find something that is not maximum so here if I start looking down I will find this edge that is not maximum so I will mark it it's not maximum and what I will do I will move it to the next one in the order I had so here I'm given an order I will move it to the next one so this will move to the next one and I erase everything I had above but I erase what I had before and to draw what I had here I will come back with the minimum path so I will choose at any step the minimum path that allows me to come back so in this case it's just that so this is the this is the successor map and if we think of this as the sequence of zeros and ones so where say this is a zero and this is one so the one in the left is a zero and the one in the right is one what we have done here is that let me go back is that here the the sequence will be one zero zero so what we did is that one zero zero something it went to sorry no I didn't have I have that one one zero zero it's okay one zero zero it went to zero one zero so this is an this can be seen as adding one in the in this daily group and what is adding one I add the coordinate and if and I keep the the how it's called the I forgot the name in English so carry here when I add one the carry yes sorry the I add the I add one here so this one becomes zero and I keep the carry here and I put it there and I continue like that so someone that if you haven't looked at this kind of dynamics if you have never looked at this then maybe you are asking yourself what happens if I never find the something that is not maximum well in that case you send to the path that is everything is a minimum of course if that is well defined if there is only one maximum and one okay but this is just to give a taste of what is a dynamics in this kind of maps this example here is what is called an odometer it's an odometer or adding machine so you have a group operation and what you're doing the successor dynamics is adding some fixed element so this is an odometer and what is the particular particular aspect of the odometer is that in each level you have one one vertex and actually that's what characterize the odometers in this context okay so and then I will just draw a few pictures that so you can imagine what happens when you have more complicated structure here the structure was quite simple just one vertex at each level but you can have much more complicated structures and the dynamics is exactly the same it's just more difficult to look at it but the idea is the same so here for instance let's consider this this path in green of course this continues down but I just draw a portion of it so the dynamics is you go down till you find something that you can move something that is not the maximum so in this case so this is maximum this is maximum and this is not maximum so I should be able to move that one so I mark that as non-the maximum and then I move it there and then I come back to the to the top following the minimum path so that's one iteration of the successor map and if your if your graph is complicated enough you will have a complicated enough dynamics so that's what I wanted to say about these maps any question not please let me know if you have any question okay so why one could one could ask after all these definitions what is the class of systems that can be built like that like that and here two important theorems are the following ones just to mention the condition that we will be using the diagram is simple if you can connect no to vertices that are far away in the diagram and this is the condition to have a minimal dynamic minimal dynamics means that every orbit is a dance is dancing in the space so say a fundamental result in this context is a theorem in the 92 by Herman Putman and Scho that show that actually any minimal counter dynamics any minimal counter system can be represented in this way so and here minimal counter system means that is a minimal topological system and this space is a counter space so this is a counter space so this is kind of a universality or represent representation theorem of all dynamics that happen in a counter system in a counter space all of them can be represented as I said before so the class of systems we can treat with these diagrams is quite big it's all counter systems and the definition we will be discussing next is the one of finite rank the one here finite rank so a system is a finite rank if I can represent it as previously with the diagram where the number of vertices doesn't increase to infinity so let me go back here here I could have that when I go more down the number of vertices can go to infinity and this is a valid system the systems of all the dynamics but in the case where this doesn't happen or when you can represent it in a way that this doesn't happen this is called a finite rank system so all systems that can be represented in this way and this class of system starts to exhibit some rigidity properties and the first one is this theorem by Tomas Nonarovich and Alejandro Mas from 2008 that says that if you have a system like this where you can or that can be represented with a diagram finite where the number of vertices is bounded then it's either expansive meaning a subshift or an odometer like the situation I draw in the beginning you have one vertex and you just you are just adding a fixed number the fixed element in that group so this is a dichotomy theorem that says everything in that class is a subshift or it's an a key continue system it's a rotation in on a compact delin group so here we have a first restriction to this class because if you are wondering counter space it can be all subshifts are happening in a counter space but you can have things that are not expansive so not everything is a subject okay so this is a very first result and this the theorem says that well more or less we can focus on on subshifts because the other case would be odometer and odometer is easier to understand and this class of systems was studied in details in 2012 by Besugli, Tiatowski, Medinet and Solomniac and they exhibited they exhibited some rigidity properties that I will briefly mention here and the first property they prove is that they have a finite number of ergodic measures finite number of ergodic measures as much as like in the case of non-superlina complexity actually the number of ergodic measures is bounded by the rank bounded less than the rank so the maybe I didn't mention the rank is like the number of vertices that you can use at each level to represent the system the minimum number and so the system have finite number of ergodic measures and they have zero entropy what they showed is that they have zero entropy for any invariant measure which of course implies that the vibrational principle that the topological entropy is zero so we have these rigidity conditions in this class which are the ones I mentioned in the beginning that happen for non-superlinear systems so here we can ask what how about the next the next restriction so let me and before presenting the next restriction I will put in context this class of systems with the the condition we had in non-superlina complexity so this is a recent part of a recent joint work with Fabian and Juran Alejandro Marz and Samurta Tit and we show that the condition of non-superlina complexity actually fits in this class meaning that if a system is like that has non-superlina complexity then it has to be a finite rank so ease of the form I described before so this gives an indirect they prove that this system have a finite number of ergodic measures for instance using the previous result I mentioned here in the this slide but it's very indirect it's not really optimal in the in the in the sense of the number so then one can ask very quickly if maybe this complexity condition characterizes this topological condition of finite rank and this is false actually even with rank two systems so this is a system that is given by two two vertices at each level in this class in a system like this you don't have you may not have a condition with the complexity so there are examples where the complexity grows more than linear so super linear complexity but is of a finite rank so just to draw a picture here we have the finite rank systems and here inside we have non-super linear this is non-super linear and this is uh these two classes are different and I mentioned in the beginning that this non-super linear class presents some restrictions like entropy number of ergodic measures which are also that also happens in the finite rank but they also have this automorphism group restrictions and asymptotic per restriction so the first question that one could ask is are those uh conditions also satisfied in finite rank system so let me show you some questions these are some questions that we post in the the paper but they are really solved so first question is does the minimal uh finite rank system have many many many finite many asymptotic components and this was solved by bastian spinosa which is a graduate student of alehandra and fabian and this this condition of sorry sorry there is a question of borisolomiaq he's asking how fast can the complexity grow for finite rank oh yeah very good question um I actually I I I wrote it down in the end so maybe we can talk about that again I I actually wrote it that's an open question I want to ask so this is um this was already solved this is I think 2020 and this of course implies that the automorphism group of these systems is also constrained and some other topological consequences or questions that is every factor of finite rank also finite rank this was not clear at all but this was also solved by uh uh let me go lestani bastiano also and go lestani I'm sorry if I pronounce it correctly and joseini this is also 2020 and um this was recently like last month solved by also by bastian spinosa that in terms of symbolic factors this class is also very restricted has finitely many of them have it has finitely many factors up to topological conjugacy and I advertise that we have here talks on on thursday so I advertise this talks that will be on thursday and I I think they will talk more in details about these results so I will just mention them I have another question and related to this all these topological questions one of the aspects I would like to treat in this class of systems is the complexity one so that that is what we are going to discuss more in details in what follows what is the complexity what is the complexity of finite rank systems we know it's a zero entropy but uh what is a possible uh complexities you can see and how how can you control them if I give you an example how how can you estimate the complexity and to do so I will introduce a few notions of morphisms and esadic subsets which are are a bit more convenient to to discuss this question so and here I will give a few more definitions I morphism will be any map I will call it say tow from an alphabet to another one that it's you can think of this as the concatenation of images of any symbols so tow of a sequence finite sequence say of a word it's just the concatenation of the the image of each letter that that's what I will call a morphism I will not call it a substitution because the alphabets may be different so to restrict the word substitution for same alphabet I will call this a morphism and given a sequence of morphisms here positive I will restrict to positive morphisms and positive means that if I write a letter say if I write tow of a I will see some letters b1 b2 etc I want to see all of them so all letters appear in the images of of the letters in the in the alphabet in the in the beginning this is the same as when you construct the matrix associated to this morphism you have a positive matrix and this is not a restrictive assumption all the systems we are looking at can be can be considered with this condition so it's not it's not anything it's not restrictive just to make things easier so we have a sequence of positive morphisms where the alphabets might be different and having that sequence of morphism we can build what is called the esadic subshift associated to the sequence of morphisms which is given by the following condition a point is in the subshift if the finite words of this point can be obtained iterating some letter from far from far away so let me try to do the picture I have a sequence of morphisms this is okay close from an even alphabet the next one and this goes to say tau zero here I have my original alphabet a0 alphabet a1 you have some way to send letters here into words here and the point in this subshift belongs to the this alphabet the first alphabet to the z and the point is in there if when I take any subword of it here I will take a subword this subword comes from iterating some letter in some say alphabet say that this is this is the ak and and you iterate this with all the morphisms and when you do that you have a word that sees this finding word other way to say it is that you start iterating morphisms from far and far away and the limit points you get are the points in the sxh so these are like a generalization of the substitution but you allow the alphabets and the morphisms to change every time so this is a way to to to build what is called the sxhift and what actually what it's proved in the recent results I mentioned is that this this is intimately related to the finite grand systems and how to relate this class of system with the finite grand systems is that finite grand systems are exactly the systems where you can control the size of the alphabet here the alphabet can grow can grow to infinity the size the size can grow to infinity but if you are if you restrict to morphisms that where the alphabet doesn't grow to infinity then you actually get the same the same class of finite grand system and this this will be explaining details I think in the talks on third because this is the it's a theorem it's not it's not it's not trivial I mean probably it's one can expect some some this type of result but there is some technical things to do in the middle okay so the next question is if I give if I give you the sequence of morphisms how can you compute the complexity or is there any way you can compute the complexity so this is the next I will spend the remaining minutes of my talk talking about this so how is how to estimate this complexity without when I give you an example or in some classes etc and well of course there are some non-case substitutions substitution is one that all the morphisms are equal to the same to given morphisms so there is only one morphism that you repeat infinitely many times and this case in this case the the the complexity sublinear meaning it's bounded it's a p of n it's bounded by constant times n another class that is well studied in in symbolic dynamics is the linearly recurrent subshift I will not give that like the original definition but this will be when the morphisms you have a finite set of finite set of morphisms so finite set of morphisms and in this case the complexity is also sublinear so it's a finite set of morphisms that you that you choose how you start iterating and then there is the finite rank which you can ask what are the complexities that are here but this is always our explanation because well it has zero entropy okay so the what we tried to develop was some ideas to compute the complexity in some examples or in some classes and I will mention you some of the tools we saw or we tried to create to to compute the complexity here everything I'm I'm talking about is minimal and the morphisms are positive okay and to understand how is the complexity we will have to look at these quantities so for a for a morphism let's denote the double bar of the morphism the the the maximal length of of tau of a letter so you write down so you have your letter a b c say you write the substitution and you may have that some letters give you a very long word some other letters give you a very short word so I will write with this the longest the longest length and with this symbol the shortest and and actually we'll see that these two quantities are behind the complexity so if you have a morphism where these two things are very different you can achieve very non-trivial complexities like high complexity okay so here there is the first tool to understand the complexity that is and is what we call the repetition complexity and okay the definition I will just write how to compute it suppose that you have a that gives you a a b c a b so what this what this complexity for this quantity means is that you will count for each letter you will count how many times you change the letter but here we started with b with a so I will start with one because I'm using a letter then here I change the letter I will plus one here I change another letter I will plus one here I change another well this is a bad example it doesn't illustrate what's happening maybe let's say that I have many b's and in dn and a so here I am not changing the letter here I change the letter I plus plus one so the this when I do this I count one two three four five six and I do that for every letter so b will give me something say seven and I add all those quantities and this is what we call the repetition complexity with how many times you can you need to change the letter in the in the morphism and that's should be related to the complexity because if you if you change too much you are creating many words somehow and that's exactly what it happens this is a computation that I will just write the like form of it but it can be very it can be done much more precisely and proposition with this quantity is the following suppose you have a sub shift then you can bound the complexity in an n or for a large enough n by so this is a it's not a precise bound but it will tell it will work to illustrate the result so you bound by the maximum of the alphabets of course this makes sense only if the I mean this is meaningful only in the case where the alphabets remain bounded and then you multiply this by the the link soup of this complexity that I defined previously so you start computing how you the morphism changed the letters and you have to multiply that by the size of the the maximum of the alphabets and this is the constant term that goes with n of course this can this can grow but this is meaningful only when this is bounded because it will give us some information when the system have sublinear complexity so if you have a static sub shift where we can bound these two things then the system has sublinear complexity sublinear complexity these are are quite strong conditions but I included it here just to illustrate how to use that notion but it can be it can be described much more precisely we can talk afterwards if you have a need and as a consequence of this result we did use some results in the equivalence strongly or with equivalence theory that says that if you have any finite topological rank then it's a strongly or with equivalent to something of sublinear complexity meaning I will just say it with words that means that you can send orbits to orbits in a somehow continuous way but this notion reserves for instance the set of infinite measures and some other under dimension group of course and this proposition gives for instance an alternative proof that the finite rank systems have finitely many agroic measures because you it's equivalent to something of sublinear complexity but of course there we lose the precise bound which is just an observation and so this is this complexity notion allows us to compute some things but actually there is more precise notion that we described that is the complexity of relative complexity of morphisms this is somehow how are how the morphisms in the aesthetic representation are being related so I think I have very few minutes so probably I will skip that part if you have if you want to take a look at this we can talk afterwards but this is a more precise notion to compute the complexity the previous one is easy to it's easy to compute in examples but it is highly non-optimal to compute the complexity this one is a bit better computing the complexity but it might be more complicated in examples but okay I will skip this because I'm running out of time okay and the consequence of these computations is that one can say when a system has zero entropy and it's it has to do with the the length the how the alphabet the exponential growth of the alphabets compared to the smallest word of the sequence of morphisms this is probably for poor result but we gave a very concise proof of this in particular finite rank have zero entropy okay so I wanted some other consequence that sub quadratic complexity is always true when you have two alphabets of size two and okay since I ran out of time let me go to the questions which I probably are more interesting so here are a few questions that are still open maybe the specialists can say if they make sense like people that have worked more on finite rank system so the first thing is if you can control in a polynomial way the complexity of a finite rank system so can it be can you find a degree that controls the polynomial the complexity of the finite rank system or maybe not and give an example where you cannot do that this is the first question so what I said previously that when the alphabet is of size two you can do it and the second question is if you can characterize complexity and having topological finite rank in some subclass of systems for instance we don't know if these conditions are equivalent in the topics class maybe in the topics class having finite topological rank is the same as having non-superlinear complexity there is some evidence that supports that but I will not mention that we can discuss it later if you have any questions and then we have a variety of other questions okay thank you Sebastian maybe we can pass to the questions there is a question in the chat room of Seijat Babel he's asking what can be saved the minimality condition is replaced by unique minimal subset in the in particular in the result of finite rank when the condition of non-superlinear complexity implies finite rankness if you have I'm not sure I I should replace I'm not sure like what happens if you drop the minimality probably you can say things but I haven't thought about it okay there is another question of Martin Lustig he's asking the chat and say what about for placing for subchips the water diversity rank by the by a nestadic rank defined through the existence of a nestadic development which is everywhere growing of buddha the level of a betranks and perhaps also recognized recognizable on each level this yeah um and you mean not not positive maybe I think actually this question of Martin will be it's one of the things that will be discussed on Thursday I think that's one is a recent paper of for Bastia and Spinoza where he addresses that question uh if I'm not mistaken I think there is some development of that other question a question of uh Dan Rust uh will you what's that the lens of each morphism is bounded what's happened in this case related to Martin uh Martin questions no they don't need to be bounded bounded they can the size can grow which will always happen if you telescope the diagram like if you look at it in the in the Bratelli version but you can you are allowed to do that okay so let's thank Sebastian
anthropological aspects of finite rank systems. I few results, then try to give some ideas on some special topics and then ask a few questions that some classes and I will mention you some of the characterize complexity and having topological