Transcendental Thurston Theory for entire functions and compositions
Shemyakov, Sergey
Thurston's topological characterization theory asks whether there is a holomorphic dynamical system that realizes topological (even combinatorial) data, this often allows to describe all possible dynamical systems in a certain parameter space. I work on extending Thurston topological characterization theory to different classes of transcendental functions. In my talk I will start with some explicit families of functions for which we have established an extension of Thurston's theory, and then describe further extensions by compositions of such functions. The presented ideas are part of my PhD thesis under the supervision of Dierk Schleicher. This is work in progress.
 Yeah, so my name is Sergei Shenekov. I finished my PhD here at X-Message University and what I will talk about today, the transcendental Thurston theory is a part of my PhD thesis and still work in progress. The history of Thurston theory starts in the 80s when William Thurston answered the question which topological models of post-created finite rational maps are realized by actual rational maps and it was nicely written in the paper of Dwardin Hubbard in 1993. And this paper I think was mentioned quite sometimes already at the conference, for example, by Dylan. And this result had quite some importance for the world of complex dynamics. For example, using Hubbard trees for a year did the classification of post-created finite polynomials as dynamical systems. Then similar classification was done for post-created finite Newton maps by Lodz, Mikulich and Leiche. And the similar result was also proved for critically fixed rational maps, the O'Helushenkant, and the Algorithma, a couple of names to mention here. And when speaking about Thurston results, I cannot skip mentioning his big vision. So he had, except for the topological classification of rational maps, he also had two other quite important, quite big results. Capabilization of three manifolds in classification of surface automorphisms. The first, about the first problem we had a nice introduction by John Hubbard on the very first lecture at the conference. And this was an important step in the geometrization problem that was finished by Grigory Perlman. And while looking at this picture, I once again want to stress the subtle importance of complex dynamics for the geometry and particular for hyperbolic geometry. Well, but I work on transcendental Thurston theory, that's what I'm interested in. The attempts to generalize the rational theory were made ever since the 80s. I would say that the milestone publication was done in 2009 by Habar-Schleich and Shishikura, exponential Thurston maps and limits of quadratic differentials. And in this paper, they answered the Thurston question for posingular finite topological exponential functions. And while many people at the conference already explained to you about the posingular, posingular finite functions, posingular finite functions, so these are the functions where the singular orbit or critical orbit is finite. And I want to mention at this point that all the functions that I will be considering in my talk will be posingular finite. And so this paper coined in a nice idea, the thick-thin decomposition for quadratic differentials, and it's quite a general tool that allowed them to tackle this transcendental case. So the next publication that I would like to mention is PhD thesis by David Sprantz in 2019. In this thesis, he generalized the theory of Habar trees to so-called homotopy Habar trees for all posingular finite entire functions. And the final publication that I want to mention, a publication by Kostya Bogdanov, infinite-dimensional Thurston theory and transcendental dynamics, which were put on dark after this year. And while it's... You will hear some more from Kostya, he will speak right after me. But you probably can notice that by looking at the schedule of our conference, that quite some talks go into the world of transcendental dynamics. And as you can see, the theory of Habar trees was promoted to the full power for entire functions. And the tools that you can see in this exponential paper, although I general enough, so I want to make a statement that transcendental Thurston theory is ready to blossom. Okay. So let's maybe get a bit more precise on what Thurston theory is. And first, you want to start with a topological model for some homomorphic function. And as an example, already it was on the conference. I have an example for topological polynomial. For example, topological polynomial is a finite-degree branch covering of a plane to a topological exponential. This is, as you can see on the slide, it's a topological cover of a plane to one-spuncher plane. So, when you have your topological model that you pick, you ask a question, can this for singularity find a topological model, have be realized by some homomorphic or meromorphic function g, depending on the class that you consider. And the way to make it precise is by Thurston equivalents. So let's have a look at the definition of the slide. If we have a topological post-singular finite function f, and then it's Thurston equivalent to some meromorphic or homomorphic function g, if we have the following diagram. So on the left-hand side, here you can see your topological function f from a sphere with punctured post-singular set to itself, to a sphere with the punctures. On the right-hand side, you see an entire meromorphic function g from Riemann's sphere with punctured post-singular set, and phi prime and phi double prime are two homomorphisms, which are isotopic relative in the set of post-singular points. Well, and let me maybe give a couple of ideas to the people who are new to Thurston theory. I won't go into too many details, but I just have to say this. The idea is a particular Thurston iteration, and you run this iteration on Taichmaner space. And the Taichmaner space, which was mentioned, for example, by Lasse and his talk, the Taichmaner space of a sphere with finitely many punctures, this would be the interest for us, is the space of conformal structures. So to be more precise, this is the space of homeomorphisms from one finitely punctured sphere to finitely punctured Riemann's sphere, modular sum, isotopic equivalence relation. And the thing that we iterate on this Taichmaner space is the Thurston pullback map. And it was already advertised by Becker this morning. But let me draw the diagram on the slides for you. So you have on the bottom of the commutative diagram, you have a phi n, which represents a point in Taichmaner space. This is some homeomorphism from punctured sphere to punctured Riemann's sphere. And on the left, you have your topological model F. So then you take the conformal structure from here, and you pull it back by the composition of phi n and F. And you uniformize this by getting the homeomorphism here, and the diagram closes up with an entire function on the right and side. And we say that phi n plus 1 is a Thurston pullback of phi n, and this is defined on Taichmaner space. So a couple of notes here. You can see from this diagram that entire functions gn plus 1 are from parameter space of topological model F. And you can think about this. If you have a topological model for exponential functions, then gn's would be from an exponential family. And also, you can see from this definition that a fixed point of this iteration will give you the solution. If you have a look here, the diagram will close up, and the function g on the right hand side would be the Thurston equivalent function. OK? OK, so what's? The map phi n plus 1 is only defined up to post composition with an automorphism of C bar. And gn plus 1, as defined, has no dynamical meaning because post composition with an automorphism of C bar will destroy any dynamical. Right. That's true. We have to fix some particular points in our family by a business. That would be more correct. So it turns out that the Thurston problem has a positive solution unless there is an invariant topological multi-curve with a particular impossible geometric mapping properties. And by topological multi-curve, I mean a collection of simple closed curves with at least two punctures, at least two points from post-singular set inside and outside. And one kind of topological obstructions that will be important for my talk is a levy cycle also, something I've already heard. And a definition of levy cycle goes like this. If f is your post-singular finite topological model, then a levy cycle is a sequence of disjoint, essential, simple, closed curves such that some component, gamma i prime of the pre-image of the curve gamma i plus 1 is homotopic to gamma i, relative to f, right? This is your cyclic behavior. And this map from the pre-image to image is a homeomorphism. So it's a one to one mapping. And the conceptual idea of my talk, if you have two entire functions that can be obstructed only with levy cycles, then the composition can also be obstructed only with levy cycles. So maybe that's a good point to ask whether you have some questions. Yes? I think as defined, well, the question was whether levy cycle is a dynamical thing or not. And Dirk has an issue with me stating that levy cycle is a dynamical object. Yes, well, this is the idea of my talk is a concept which I don't want to define precisely. This is just a taste of what you will see. So I don't want to digest this couple of lines on the bottom of the slide too precisely because it will require me some talking and to get into the things that I don't want to get in my 30 minutes that I have for presentation. OK. That's good. So first, I would like to speak about quadratic differentials. Quaradic differentials were the powerhouse of the exponential paper that I mentioned. And the reason for this is that the space of finite norm mirror morphic quadratic differentials is cotangent to the tight mirror space. And this was beautifully shown by John Hubbard on the first talk. Well, and why is this important for us? It's important because contraction in the cotangent space implies convergence in tight mirror space. And maybe I can make an analogy here with something that we all are familiar with. If you have a rubber duck that swims on a D minus two dimensional pond and you push it with a force that is ever decreasing, then at some point your duck will stop. And this is quite similar to what we do, but we do it in discrete time. So a conceptual take away of quadratic differentials is that they help to split the problem into functional question and dynamical question. It will be in how a particular function pushes forward the quadratic differential. And dynamical meaning what combining those push forwards along the orbit, what do they tell us about the dynamics of the function? And let me get a bit more precise. So a mirror morphic quadratic differential q is an object on a Riemann surface that locally looks like q of z, dz square, and this gives you the coordinate change between the charts on the intersect. q is a mirror morphic function. And an important notion for me that I will mention a lot is the norm of quadratic differential, which I will be calling mass from now on. If we have a quadratic differential q, then the norm on some domain x would be the following integral, norm of qz, norm of dz squared, this was expressed by John Dexdy. And a couple of properties of mass that will be important for me. That's the first mass is a zero on open domains, but also mass can concentrate on small domains. So for example, if you have a double pole of q, then the mass around this double pole will be infinite. And if you have a couple of simple poles that come together, then the mass on the small disks around them will be very big, and it will decrease exponentially away from the poles, at least in some neighborhood. Okay. And I also want to say about push forward of quadratic differential by polymorphic or mirror morphic function. And a push forward is just a quadratic differential that is defined on the image. Even surface. An important property for me would be this property with the composition. So the push forward by a composition splits into two consecutive push forwards. And also I will be speaking about how push forwards of quadratic differential by functions affect mass of quadratic differential. Push forwards by injective functions preserve mass, but usually some masses cancel. This means that this ratio that you can see here of mass of pushed forward differential to the original differential is usually less than one. And I will say mass cancellation that would mean one minus this factor, and the estimating mass cancellation is the main work which was done in the exponential paper. So maybe again, other questions about quadratic differential? Okay. So once again. So I'm wondering if the push forward operator is always well defined in this case. Yes. So infinite to one mapping still is perfectly fine with push forwards. The thing is the push forward of quadratic differential by function may have many inverse branches. So you will have a couple of domains that are mapped on top of each other. And their quadratic differential will interfere in some way, but it's still perfectly defined. Okay. So maybe let me move closer to the main point of my talk. The first definition that I have to give is a definition of a Levy disk. Again this is not a dynamical definition. Let m be some modulus and g be an entire function and p be a finite set in a complex plane. And you might think about p as the more singular set that we saw earlier. So let a be an annulus of modulus m, like on the picture, and c be its bounded complementary component. And we denote d as a pair AC or sometimes we abuse the notation and we say it's a union. So the d is an M Levy disk for function g. If g is by holomorphic on the union of a and c. And a is essential with respect to mark points p. That means that there are at least two points from this set e inside and outside of the annulus a. So once again, the important thing is about Levy disks. They have modulus m. Function is by holomorphic, so it's one to one on Levy disk. And the annulus is essential with respect to our mark point. This is a Levy disk and the next definition of Levy type functions, perhaps the most important definition in my talk. So I'll give it on various levels of details. So Levy type functions are the functions where mass can survive the push forward without consolation only on Levy disk. So being a bit more precise, if the mass loss in push forward of q by a Levy type function and g is small, then almost all mass of q is located on Levy disks for g of big modulus. And big modulus is important. So now to the precise definition of Levy type function, let p be a finite set. And again, you may think about the singular set and g be an entire function. Then we say that g is Levy type. If for any quadratic differential, q with poles in p, any modulus m, any fraction of mass outside disk epsilon, there is a fraction of preserved mass eta such that once the mass constellation is small, so the fraction of preserved mass is bigger than eta. Then there are m Levy disks for function g of modulus bigger or equal than m such that all but epsilon mass sits on these disks. And this definition might sound a bit technical, but in fact, in the exponential paper, it was proven that exponential family of functions are Levy type functions. And another result that we have proven recently is that these multi-era functions are also Levy type functions. And if you have a look at the formula, it doesn't tell you too much. But in fact, functions of this kind, where p is a polynomial, are the functions with finally many acycotic tracks, and that's why they have interest to us and they really have some meaning. And why Levy type functions are so nice is because of the Levy disk principle. And Levy disk principle tells us, if you have some topological post-singular function f, then you assume that all of the entire functions in its parameter space are Levy type. Then such topological model is realized unless there is a Levy cycle. So Levy abstractions are only possible abstractions for function f, for which all the entire functions are Levy type. And here I have to apologize because it's the only slide in my presentation that requires some detailed knowledge of person iteration. But I want to say a couple of words about the proof, about the ideas of the proof of Levy disk principle. All right, so assume that our function is not realizable. And this necessarily means that there is an efficient, so-called efficient sequence of functions gn and quadratic differential qn along the orbit of our iteration in tight mirror space. And efficient means that the ratio of mass cancellation, well, this ratio goes to 1. And functions gn are Levy type, and this means that far enough along the orbit almost all mass sits on Levy disks, like you can see here on the picture. And I hope people online can switch from the slides to the blackboard to see what I have drawn here. You hear a bunch of Levy disks for function gn, and we know that almost all mass sits on them, and they have big modulus. And we can map them forward by function gn into some other moduli, any line of big moduli. And when we do this sufficiently many times, we map them around, and it necessarily has to happen for some Levy disks like this, that in a finite number of iterations that depends only on combinatorics of the function, it travels around, and its image comes back into its homotopy class. OK, so this means that the core curve of this annulus has the cyclic behavior, and if you remember the definition of a Levy disk, the biomorphicity of the map on it was a key point in the definition. So this means that all of the functions between core curves will be one to one, and this red curves on the picture give you a Levy cycle. So this is the proof, if the function is not realizable, then you get your Levy cycle. OK, so the... So you're claiming here a theorem which 50 people have worked on 20 years, and I feel you were a little swift. OK. I didn't actually manage to understand your definition of a Levy disk. I'd like you to go back to it. I mean, if you could actually justify that that's theorem is true, well, you have a fields metal coming to you. That's an interesting statement. I mean, it's not something to dismiss by just one slide. Well... I would really love to go into the details of the proof and explain them to you. Unfortunately, I don't think that I have time for this right now, and also I'm pretty sure that some of the people in the audience won't be able to really follow the discussion, but I definitely would be happy to convince you that this principle, which I don't claim to be proven at the moment, that this principle is true and I'm really going with the fields metal. OK. Yeah, but let me maybe quickly... Part one, we've all known... Yes. Well, my claim would be that the Levy disk principle is somehow... You can already read this through the lines in the exponential paper at this moment. So I don't think that this is such a groundbreaking thing, but again, let's maybe discuss it a bit afterwards, because if the Levy principle is a dynamic type statement, then the following theorem that actually came to be a theorem is a functional statement. It does not involve any dynamics, and it tells you if F and G are two Levy type functions, then the composition is also Levy type. Well, and we're kind of... I gather that the condition that what makes this not absolutely revolutionary is the hypothesis that all the functions G and Levy type. Right. Proving that all the functions of Levy type is really all the work that has to be done in order to put this to practice. Okay, so that's really the hypothesis that makes this theorem. Right. Okay. Okay. Yeah, so let us get back to our functional statement. And I was saying that we already kind of used this theorem as an idea for the Levy disk principle. So assume that the mass constellation for the composition is small, and then the mass constellation for each of the functions must be small due to the properties of push forwards that I mentioned earlier. So you take the Levy disks, you take the Levy disks for function G, and you map them forward, and the disks, well, kind of you can see it on this picture, the images of the disks will be almost the Levy disks for the next function F, just because they share almost all mass of correct differential Q. And then you have actually to think about a bit about the geometry of what's going on, and you have to tweak the mass constellation ratio. But in the end, you can get arbitrary big modulus and arbitrarily small fraction of mass outside of the Levy disk for the intersection of these images of Levy disks for G and Levy disks for F. So this is also, I wouldn't call this theorem a ground breaking result. But what do the Levy disk principle and the Levy composition, the Levy disk composition theorem seem to imply is that if you have two functional families for which, for each of them, Levy abstraction are the only possible type of abstraction, then for the composition, also Levy abstraction should be only possible type of abstraction, and your topological model should be realizable otherwise. So where do we go from here? One goal is to put more functions in the Levy type. This is perhaps what John would be excited to hear about. And I also work on this in my PGD thesis. So this will be in the thesis as well. There is some more work coming this direction. So the next thing, you can actually add critical points to the framework of Levy disks. If you remember, Levy disks mapping one to one was an important piece of definition, but you can actually relax it. You can allow for branched mappings, and you can develop the theory of so-called branch disks and branch disk principle. And for example, this helped me to get a proof for two-genometric family, which as you can see is a composition of a function with two critical points with exponential Levy type function. And maybe a small remark here. You see that the rational function has a pole, but the composition is entire. And I work with entire functions. It would be sufficient for me to have an entire composition. Well, in the final, and perhaps the most ambitious goal for me is to add the poles into the framework of Levy disks. So this would require quite some work because we should see all the other possible obstructions, not only Levy cycles, but all the other obstructions. They come with poles. And this would require understanding the paper, the rational paper, by Duodin Hubbard well enough. So this is a challenging task. But at this point, we believe that this should be doable. So I hope I told you some interesting stuff about transcendental Thurston theory and even for people who are not really aware about Thurston theory too much before this conference. And thank you for your attention.
 Yeah, so my name is Sergei Shenekov.  I finished my PhD here at X-Message University and what I will talk about today, the transcendental  Thurston theory is a part of my PhD thesis and still work in progress.  The history of Thurston theory starts in the 80s when William Thurston answered the question  which topological models of post-created finite rational maps are realized by actual rational  maps and it was nicely written in the paper of Dwardin Hubbard in 1993.  And this paper I think was mentioned quite sometimes already at the conference, for example,  by Dylan.  And this result had quite some importance for the world of complex dynamics.  For example, using Hubbard trees for a year did the classification of post-created finite  polynomials as dynamical systems.  Then similar classification was done for post-created finite Newton maps by Lodz, Mikulich and Leiche.  And the similar result was also proved for critically fixed rational maps, the O'Helushenkant,  and the Algorithma, a couple of names to mention here.  And when speaking about Thurston results, I cannot skip mentioning his big vision.  So he had, except for the topological classification of rational maps, he also had two other quite  important, quite big results.  Capabilization of three manifolds in classification of surface automorphisms.  The first, about the first problem we had a nice introduction by John Hubbard on the  very first lecture at the conference.  And this was an important step in the geometrization problem that was finished by Grigory Perlman.  And while looking at this picture, I once again want to stress the subtle importance  of complex dynamics for the geometry and particular for hyperbolic geometry.  Well, but I work on transcendental Thurston theory, that's what I'm interested in.  The attempts to generalize the rational theory were made ever since the 80s.  I would say that the milestone publication was done in 2009 by Habar-Schleich and Shishikura,  exponential Thurston maps and limits of quadratic differentials.  And in this paper, they answered the Thurston question for posingular finite topological  exponential functions.  And while many people at the conference already explained to you about the posingular, posingular  finite functions, posingular finite functions, so these are the functions where the singular  orbit or critical orbit is finite.  And I want to mention at this point that all the functions that I will be considering  in my talk will be posingular finite.  And so this paper coined in a nice idea, the thick-thin decomposition for quadratic  differentials, and it's quite a general tool that allowed them to tackle this transcendental  case.  So the next publication that I would like to mention is PhD thesis by David Sprantz  in 2019.  In this thesis, he generalized the theory of Habar trees to so-called homotopy Habar  trees for all posingular finite entire functions.  And the final publication that I want to mention, a publication by Kostya Bogdanov,  infinite-dimensional Thurston theory and transcendental dynamics, which were put on  dark after this year.  And while it's...  You will hear some more from Kostya, he will speak right after me.  But you probably can notice that by looking at the schedule of our conference, that quite  some talks go into the world of transcendental dynamics.  And as you can see, the theory of Habar trees was promoted to the full power for entire  functions.  And the tools that you can see in this exponential paper, although I general enough, so I want  to make a statement that transcendental Thurston theory is ready to blossom.  Okay.  So let's maybe get a bit more precise on what Thurston theory is.  And first, you want to start with a topological model for some homomorphic function.  And as an example, already it was on the conference.  I have an example for topological polynomial.  For example, topological polynomial is a finite-degree branch covering of a plane to  a topological exponential.  This is, as you can see on the slide, it's a topological cover of a plane to one-spuncher  plane.  So, when you have your topological model that you pick, you ask a question, can this  for singularity find a topological model, have be realized by some homomorphic or meromorphic  function g, depending on the class that you consider.  And the way to make it precise is by Thurston equivalents.  So let's have a look at the definition of the slide.  If we have a topological post-singular finite function f, and then it's Thurston equivalent  to some meromorphic or homomorphic function g, if we have the following diagram.  So on the left-hand side, here you can see your topological function f from a sphere  with punctured post-singular set to itself, to a sphere with the punctures.  On the right-hand side, you see an entire meromorphic function g from Riemann's sphere  with punctured post-singular set, and phi prime and phi double prime are two homomorphisms,  which are isotopic relative in the set of post-singular points.  Well, and let me maybe give a couple of ideas to the people who are new to Thurston theory.  I won't go into too many details, but I just have to say this.  The idea is a particular Thurston iteration, and you run this iteration on Taichmaner space.  And the Taichmaner space, which was mentioned, for example, by Lasse and his talk, the Taichmaner  space of a sphere with finitely many punctures, this would be the interest for us, is the  space of conformal structures.  So to be more precise, this is the space of homeomorphisms from one finitely punctured  sphere to finitely punctured Riemann's sphere, modular sum, isotopic equivalence relation.  And the thing that we iterate on this Taichmaner space is the Thurston pullback map.  And it was already advertised by Becker this morning.  But let me draw the diagram on the slides for you.  So you have on the bottom of the commutative diagram, you have a phi n, which represents  a point in Taichmaner space.  This is some homeomorphism from punctured sphere to punctured Riemann's sphere.  And on the left, you have your topological model F. So then you take the conformal structure  from here, and you pull it back by the composition of phi n and F. And you uniformize this by  getting the homeomorphism here, and the diagram closes up with an entire function on the right  and side.  And we say that phi n plus 1 is a Thurston pullback of phi n, and this is defined on  Taichmaner space.  So a couple of notes here.  You can see from this diagram that entire functions gn plus 1 are from parameter space  of topological model F. And you can think about this.  If you have a topological model for exponential functions, then gn's would be from an exponential  family.  And also, you can see from this definition that a fixed point of this iteration will  give you the solution.  If you have a look here, the diagram will close up, and the function g on the right  hand side would be the Thurston equivalent function.  OK?  OK, so what's?  The map phi n plus 1 is only defined up to post composition with an automorphism of C  bar.  And gn plus 1, as defined, has no dynamical meaning because post composition with an automorphism  of C bar will destroy any dynamical.  Right.  That's true.  We have to fix some particular points in our family by a business.  That would be more correct.  So it turns out that the Thurston problem has a positive solution unless there is an invariant  topological multi-curve with a particular impossible geometric mapping properties.  And by topological multi-curve, I mean a collection of simple closed curves with at  least two punctures, at least two points from post-singular set inside and outside.  And one kind of topological obstructions that will be important for my talk is a levy cycle  also, something I've already heard.  And a definition of levy cycle goes like this.  If f is your post-singular finite topological model, then a levy cycle is a sequence of  disjoint, essential, simple, closed curves such that some component, gamma i prime of  the pre-image of the curve gamma i plus 1 is homotopic to gamma i, relative to f, right?  This is your cyclic behavior.  And this map from the pre-image to image is a homeomorphism.  So it's a one to one mapping.  And the conceptual idea of my talk, if you have two entire functions that can be obstructed  only with levy cycles, then the composition can also be obstructed only with levy cycles.  So maybe that's a good point to ask whether you have some questions.  Yes?  I think as defined, well, the question was whether levy cycle is a dynamical thing or  not.  And Dirk has an issue with me stating that levy cycle is a dynamical object.  Yes, well, this is the idea of my talk is a concept which I don't want to define precisely.  This is just a taste of what you will see.  So I don't want to digest this couple of lines on the bottom of the slide too precisely  because it will require me some talking and to get into the things that I don't want to  get in my 30 minutes that I have for presentation.  OK.  That's good.  So first, I would like to speak about quadratic differentials.  Quaradic differentials were the powerhouse of the exponential paper that I mentioned.  And the reason for this is that the space of finite norm mirror morphic quadratic differentials  is cotangent to the tight mirror space.  And this was beautifully shown by John Hubbard on the first talk.  Well, and why is this important for us?  It's important because contraction in the cotangent space implies convergence in tight  mirror space.  And maybe I can make an analogy here with something that we all are familiar with.  If you have a rubber duck that swims on a D minus two dimensional pond and you push it  with a force that is ever decreasing, then at some point your duck will stop.  And this is quite similar to what we do, but we do it in discrete time.  So a conceptual take away of quadratic differentials is that they help to split the problem into  functional question and dynamical question.  It will be in how a particular function pushes forward the quadratic differential.  And dynamical meaning what combining those push forwards along the orbit, what do they  tell us about the dynamics of the function?  And let me get a bit more precise.  So a mirror morphic quadratic differential q is an object on a Riemann surface that locally  looks like q of z, dz square, and this gives you the coordinate change between the charts  on the intersect.  q is a mirror morphic function.  And an important notion for me that I will mention a lot is the norm of quadratic differential,  which I will be calling mass from now on.  If we have a quadratic differential q, then the norm on some domain x would be the following  integral, norm of qz, norm of dz squared, this was expressed by John Dexdy.  And a couple of properties of mass that will be important for me.  That's the first mass is a zero on open domains, but also mass can concentrate on small domains.  So for example, if you have a double pole of q, then the mass around this double pole  will be infinite.  And if you have a couple of simple poles that come together, then the mass on the small  disks around them will be very big, and it will decrease exponentially away from the  poles, at least in some neighborhood.  Okay.  And I also want to say about push forward of quadratic differential by polymorphic or  mirror morphic function.  And a push forward is just a quadratic differential that is defined on the image.  Even surface.  An important property for me would be this property with the composition.  So the push forward by a composition splits into two consecutive push forwards.  And also I will be speaking about how push forwards of quadratic differential by functions  affect mass of quadratic differential.  Push forwards by injective functions preserve mass, but usually some masses cancel.  This means that this ratio that you can see here of mass of pushed forward differential  to the original differential is usually less than one.  And I will say mass cancellation that would mean one minus this factor, and the estimating  mass cancellation is the main work which was done in the exponential paper.  So maybe again, other questions about quadratic differential?  Okay.  So once again.  So I'm wondering if the push forward operator is always well defined in this case.  Yes.  So infinite to one mapping still is perfectly fine with push forwards.  The thing is the push forward of quadratic differential by function may have many inverse  branches.  So you will have a couple of domains that are mapped on top of each other.  And their quadratic differential will interfere in some way, but it's still perfectly defined.  Okay.  So maybe let me move closer to the main point of my talk.  The first definition that I have to give is a definition of a Levy disk.  Again this is not a dynamical definition.  Let m be some modulus and g be an entire function and p be a finite set in a complex  plane.  And you might think about p as the more singular set that we saw earlier.  So let a be an annulus of modulus m, like on the picture, and c be its bounded complementary  component.  And we denote d as a pair AC or sometimes we abuse the notation and we say it's a union.  So the d is an M Levy disk for function g.  If g is by holomorphic on the union of a and c.  And a is essential with respect to mark points p.  That means that there are at least two points from this set e inside and outside of the  annulus a.  So once again, the important thing is about Levy disks.  They have modulus m.  Function is by holomorphic, so it's one to one on Levy disk.  And the annulus is essential with respect to our mark point.  This is a Levy disk and the next definition of Levy type functions, perhaps the most important  definition in my talk.  So I'll give it on various levels of details.  So Levy type functions are the functions where mass can survive the push forward without  consolation only on Levy disk.  So being a bit more precise, if the mass loss in push forward of q by a Levy type function  and g is small, then almost all mass of q is located on Levy disks for g of big modulus.  And big modulus is important.  So now to the precise definition of Levy type function, let p be a finite set.  And again, you may think about the singular set and g be an entire function.  Then we say that g is Levy type.  If for any quadratic differential, q with poles in p, any modulus m, any fraction of  mass outside disk epsilon, there is a fraction of preserved mass eta such that once the mass  constellation is small, so the fraction of preserved mass is bigger than eta.  Then there are m Levy disks for function g of modulus bigger or equal than m such that  all but epsilon mass sits on these disks.  And this definition might sound a bit technical, but in fact, in the exponential paper, it  was proven that exponential family of functions are Levy type functions.  And another result that we have proven recently is that these multi-era functions are also  Levy type functions.  And if you have a look at the formula, it doesn't tell you too much.  But in fact, functions of this kind, where p is a polynomial, are the functions with  finally many acycotic tracks, and that's why they have interest to us and they really  have some meaning.  And why Levy type functions are so nice is because of the Levy disk principle.  And Levy disk principle tells us, if you have some topological post-singular function f,  then you assume that all of the entire functions in its parameter space are Levy type.  Then such topological model is realized unless there is a Levy cycle.  So Levy abstractions are only possible abstractions for function f, for which all the entire functions  are Levy type.  And here I have to apologize because it's the only slide in my presentation that requires  some detailed knowledge of person iteration.  But I want to say a couple of words about the proof, about the ideas of the proof of  Levy disk principle.  All right, so assume that our function is not realizable.  And this necessarily means that there is an efficient, so-called efficient sequence of  functions gn and quadratic differential qn along the orbit of our iteration in tight  mirror space.  And efficient means that the ratio of mass cancellation, well, this ratio goes to 1.  And functions gn are Levy type, and this means that far enough along the orbit almost all  mass sits on Levy disks, like you can see here on the picture.  And I hope people online can switch from the slides to the blackboard to see what I have  drawn here.  You hear a bunch of Levy disks for function gn, and we know that almost all mass sits  on them, and they have big modulus.  And we can map them forward by function gn into some other moduli, any line of big moduli.  And when we do this sufficiently many times, we map them around, and it necessarily has  to happen for some Levy disks like this, that in a finite number of iterations that depends  only on combinatorics of the function, it travels around, and its image comes back into  its homotopy class.  OK, so this means that the core curve of this annulus has the cyclic behavior, and if you  remember the definition of a Levy disk, the biomorphicity of the map on it was a key point  in the definition.  So this means that all of the functions between core curves will be one to one, and this red  curves on the picture give you a Levy cycle.  So this is the proof, if the function is not realizable, then you get your Levy cycle.  OK, so the...  So you're claiming here a theorem which 50 people have worked on 20 years, and I feel  you were a little swift.  OK.  I didn't actually manage to understand your definition of a Levy disk.  I'd like you to go back to it.  I mean, if you could actually justify that that's theorem is true, well, you have a fields  metal coming to you.  That's an interesting statement.  I mean, it's not something to dismiss by just one slide.  Well...  I would really love to go into the details of the proof and explain them to you.  Unfortunately, I don't think that I have time for this right now, and also I'm pretty sure  that some of the people in the audience won't be able to really follow the discussion, but  I definitely would be happy to convince you that this principle, which I don't claim to  be proven at the moment, that this principle is true and I'm really going with the fields  metal.  OK.  Yeah, but let me maybe quickly...  Part one, we've all known...  Yes.  Well, my claim would be that the Levy disk principle is somehow...  You can already read this through the lines in the exponential paper at this moment.  So I don't think that this is such a groundbreaking thing, but again, let's maybe discuss it  a bit afterwards, because if the Levy principle is a dynamic type statement, then the following  theorem that actually came to be a theorem is a functional statement.  It does not involve any dynamics, and it tells you if F and G are two Levy type functions,  then the composition is also Levy type.  Well, and we're kind of...  I gather that the condition that what makes this not absolutely revolutionary is the hypothesis  that all the functions G and Levy type.  Right.  Proving that all the functions of Levy type is really all the work that has to be done  in order to put this to practice.  Okay, so that's really the hypothesis that makes this theorem.  Right.  Okay.  Okay.  Yeah, so let us get back to our functional statement.  And I was saying that we already kind of used this theorem as an idea for the Levy disk  principle.  So assume that the mass constellation for the composition is small, and then the mass  constellation for each of the functions must be small due to the properties of push forwards  that I mentioned earlier.  So you take the Levy disks, you take the Levy disks for function G, and you map them  forward, and the disks, well, kind of you can see it on this picture, the images of  the disks will be almost the Levy disks for the next function F, just because they share  almost all mass of correct differential Q.  And then you have actually to think about a bit about the geometry of what's going on,  and you have to tweak the mass constellation ratio.  But in the end, you can get arbitrary big modulus and arbitrarily small fraction of mass  outside of the Levy disk for the intersection of these images of Levy disks for G and Levy  disks for F.  So this is also, I wouldn't call this theorem a ground breaking result.  But what do the Levy disk principle and the Levy composition, the Levy disk composition  theorem seem to imply is that if you have two functional families for which, for each  of them, Levy abstraction are the only possible type of abstraction, then for the composition,  also Levy abstraction should be only possible type of abstraction, and your topological  model should be realizable otherwise.  So where do we go from here?  One goal is to put more functions in the Levy type.  This is perhaps what John would be excited to hear about.  And I also work on this in my PGD thesis.  So this will be in the thesis as well.  There is some more work coming this direction.  So the next thing, you can actually add critical points to the framework of Levy disks.  If you remember, Levy disks mapping one to one was an important piece of definition, but  you can actually relax it.  You can allow for branched mappings, and you can develop the theory of so-called branch  disks and branch disk principle.  And for example, this helped me to get a proof for two-genometric family, which as you can  see is a composition of a function with two critical points with exponential Levy type  function.  And maybe a small remark here.  You see that the rational function has a pole, but the composition is entire.  And I work with entire functions.  It would be sufficient for me to have an entire composition.  Well, in the final, and perhaps the most ambitious goal for me is to add the poles into the framework  of Levy disks.  So this would require quite some work because we should see all the other possible obstructions,  not only Levy cycles, but all the other obstructions.  They come with poles.  And this would require understanding the paper, the rational paper, by Duodin Hubbard well  enough.  So this is a challenging task.  But at this point, we believe that this should be doable.  So I hope I told you some interesting stuff about transcendental Thurston theory and even  for people who are not really aware about Thurston theory too much before this conference.  And thank you for your attention.
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Yeah, so my name is Sergei Shenekov. I finished my PhD here at X-Message University and what I will talk about today, the transcendental Thurston theory is a part of my PhD thesis and still work in progress. The history of Thurston theory starts in the 80s when William Thurston answered the question which topological models of post-created finite rational maps are realized by actual rational maps and it was nicely written in the paper of Dwardin Hubbard in 1993. And this paper I think was mentioned quite sometimes already at the conference, for example, by Dylan. And this result had quite some importance for the world of complex dynamics. For example, using Hubbard trees for a year did the classification of post-created finite polynomials as dynamical systems. Then similar classification was done for post-created finite Newton maps by Lodz, Mikulich and Leiche. And the similar result was also proved for critically fixed rational maps, the O'Helushenkant, and the Algorithma, a couple of names to mention here. And when speaking about Thurston results, I cannot skip mentioning his big vision. So he had, except for the topological classification of rational maps, he also had two other quite important, quite big results. Capabilization of three manifolds in classification of surface automorphisms. The first, about the first problem we had a nice introduction by John Hubbard on the very first lecture at the conference. And this was an important step in the geometrization problem that was finished by Grigory Perlman. And while looking at this picture, I once again want to stress the subtle importance of complex dynamics for the geometry and particular for hyperbolic geometry. Well, but I work on transcendental Thurston theory, that's what I'm interested in. The attempts to generalize the rational theory were made ever since the 80s. I would say that the milestone publication was done in 2009 by Habar-Schleich and Shishikura, exponential Thurston maps and limits of quadratic differentials. And in this paper, they answered the Thurston question for posingular finite topological exponential functions. And while many people at the conference already explained to you about the posingular, posingular finite functions, posingular finite functions, so these are the functions where the singular orbit or critical orbit is finite. And I want to mention at this point that all the functions that I will be considering in my talk will be posingular finite. And so this paper coined in a nice idea, the thick-thin decomposition for quadratic differentials, and it's quite a general tool that allowed them to tackle this transcendental case. So the next publication that I would like to mention is PhD thesis by David Sprantz in 2019. In this thesis, he generalized the theory of Habar trees to so-called homotopy Habar trees for all posingular finite entire functions. And the final publication that I want to mention, a publication by Kostya Bogdanov, infinite-dimensional Thurston theory and transcendental dynamics, which were put on dark after this year. And while it's... You will hear some more from Kostya, he will speak right after me. But you probably can notice that by looking at the schedule of our conference, that quite some talks go into the world of transcendental dynamics. And as you can see, the theory of Habar trees was promoted to the full power for entire functions. And the tools that you can see in this exponential paper, although I general enough, so I want to make a statement that transcendental Thurston theory is ready to blossom. Okay. So let's maybe get a bit more precise on what Thurston theory is. And first, you want to start with a topological model for some homomorphic function. And as an example, already it was on the conference. I have an example for topological polynomial. For example, topological polynomial is a finite-degree branch covering of a plane to a topological exponential. This is, as you can see on the slide, it's a topological cover of a plane to one-spuncher plane. So, when you have your topological model that you pick, you ask a question, can this for singularity find a topological model, have be realized by some homomorphic or meromorphic function g, depending on the class that you consider. And the way to make it precise is by Thurston equivalents. So let's have a look at the definition of the slide. If we have a topological post-singular finite function f, and then it's Thurston equivalent to some meromorphic or homomorphic function g, if we have the following diagram. So on the left-hand side, here you can see your topological function f from a sphere with punctured post-singular set to itself, to a sphere with the punctures. On the right-hand side, you see an entire meromorphic function g from Riemann's sphere with punctured post-singular set, and phi prime and phi double prime are two homomorphisms, which are isotopic relative in the set of post-singular points. Well, and let me maybe give a couple of ideas to the people who are new to Thurston theory. I won't go into too many details, but I just have to say this. The idea is a particular Thurston iteration, and you run this iteration on Taichmaner space. And the Taichmaner space, which was mentioned, for example, by Lasse and his talk, the Taichmaner space of a sphere with finitely many punctures, this would be the interest for us, is the space of conformal structures. So to be more precise, this is the space of homeomorphisms from one finitely punctured sphere to finitely punctured Riemann's sphere, modular sum, isotopic equivalence relation. And the thing that we iterate on this Taichmaner space is the Thurston pullback map. And it was already advertised by Becker this morning. But let me draw the diagram on the slides for you. So you have on the bottom of the commutative diagram, you have a phi n, which represents a point in Taichmaner space. This is some homeomorphism from punctured sphere to punctured Riemann's sphere. And on the left, you have your topological model F. So then you take the conformal structure from here, and you pull it back by the composition of phi n and F. And you uniformize this by getting the homeomorphism here, and the diagram closes up with an entire function on the right and side. And we say that phi n plus 1 is a Thurston pullback of phi n, and this is defined on Taichmaner space. So a couple of notes here. You can see from this diagram that entire functions gn plus 1 are from parameter space of topological model F. And you can think about this. If you have a topological model for exponential functions, then gn's would be from an exponential family. And also, you can see from this definition that a fixed point of this iteration will give you the solution. If you have a look here, the diagram will close up, and the function g on the right hand side would be the Thurston equivalent function. OK? OK, so what's? The map phi n plus 1 is only defined up to post composition with an automorphism of C bar. And gn plus 1, as defined, has no dynamical meaning because post composition with an automorphism of C bar will destroy any dynamical. Right. That's true. We have to fix some particular points in our family by a business. That would be more correct. So it turns out that the Thurston problem has a positive solution unless there is an invariant topological multi-curve with a particular impossible geometric mapping properties. And by topological multi-curve, I mean a collection of simple closed curves with at least two punctures, at least two points from post-singular set inside and outside. And one kind of topological obstructions that will be important for my talk is a levy cycle also, something I've already heard. And a definition of levy cycle goes like this. If f is your post-singular finite topological model, then a levy cycle is a sequence of disjoint, essential, simple, closed curves such that some component, gamma i prime of the pre-image of the curve gamma i plus 1 is homotopic to gamma i, relative to f, right? This is your cyclic behavior. And this map from the pre-image to image is a homeomorphism. So it's a one to one mapping. And the conceptual idea of my talk, if you have two entire functions that can be obstructed only with levy cycles, then the composition can also be obstructed only with levy cycles. So maybe that's a good point to ask whether you have some questions. Yes? I think as defined, well, the question was whether levy cycle is a dynamical thing or not. And Dirk has an issue with me stating that levy cycle is a dynamical object. Yes, well, this is the idea of my talk is a concept which I don't want to define precisely. This is just a taste of what you will see. So I don't want to digest this couple of lines on the bottom of the slide too precisely because it will require me some talking and to get into the things that I don't want to get in my 30 minutes that I have for presentation. OK. That's good. So first, I would like to speak about quadratic differentials. Quaradic differentials were the powerhouse of the exponential paper that I mentioned. And the reason for this is that the space of finite norm mirror morphic quadratic differentials is cotangent to the tight mirror space. And this was beautifully shown by John Hubbard on the first talk. Well, and why is this important for us? It's important because contraction in the cotangent space implies convergence in tight mirror space. And maybe I can make an analogy here with something that we all are familiar with. If you have a rubber duck that swims on a D minus two dimensional pond and you push it with a force that is ever decreasing, then at some point your duck will stop. And this is quite similar to what we do, but we do it in discrete time. So a conceptual take away of quadratic differentials is that they help to split the problem into functional question and dynamical question. It will be in how a particular function pushes forward the quadratic differential. And dynamical meaning what combining those push forwards along the orbit, what do they tell us about the dynamics of the function? And let me get a bit more precise. So a mirror morphic quadratic differential q is an object on a Riemann surface that locally looks like q of z, dz square, and this gives you the coordinate change between the charts on the intersect. q is a mirror morphic function. And an important notion for me that I will mention a lot is the norm of quadratic differential, which I will be calling mass from now on. If we have a quadratic differential q, then the norm on some domain x would be the following integral, norm of qz, norm of dz squared, this was expressed by John Dexdy. And a couple of properties of mass that will be important for me. That's the first mass is a zero on open domains, but also mass can concentrate on small domains. So for example, if you have a double pole of q, then the mass around this double pole will be infinite. And if you have a couple of simple poles that come together, then the mass on the small disks around them will be very big, and it will decrease exponentially away from the poles, at least in some neighborhood. Okay. And I also want to say about push forward of quadratic differential by polymorphic or mirror morphic function. And a push forward is just a quadratic differential that is defined on the image. Even surface. An important property for me would be this property with the composition. So the push forward by a composition splits into two consecutive push forwards. And also I will be speaking about how push forwards of quadratic differential by functions affect mass of quadratic differential. Push forwards by injective functions preserve mass, but usually some masses cancel. This means that this ratio that you can see here of mass of pushed forward differential to the original differential is usually less than one. And I will say mass cancellation that would mean one minus this factor, and the estimating mass cancellation is the main work which was done in the exponential paper. So maybe again, other questions about quadratic differential? Okay. So once again. So I'm wondering if the push forward operator is always well defined in this case. Yes. So infinite to one mapping still is perfectly fine with push forwards. The thing is the push forward of quadratic differential by function may have many inverse branches. So you will have a couple of domains that are mapped on top of each other. And their quadratic differential will interfere in some way, but it's still perfectly defined. Okay. So maybe let me move closer to the main point of my talk. The first definition that I have to give is a definition of a Levy disk. Again this is not a dynamical definition. Let m be some modulus and g be an entire function and p be a finite set in a complex plane. And you might think about p as the more singular set that we saw earlier. So let a be an annulus of modulus m, like on the picture, and c be its bounded complementary component. And we denote d as a pair AC or sometimes we abuse the notation and we say it's a union. So the d is an M Levy disk for function g. If g is by holomorphic on the union of a and c. And a is essential with respect to mark points p. That means that there are at least two points from this set e inside and outside of the annulus a. So once again, the important thing is about Levy disks. They have modulus m. Function is by holomorphic, so it's one to one on Levy disk. And the annulus is essential with respect to our mark point. This is a Levy disk and the next definition of Levy type functions, perhaps the most important definition in my talk. So I'll give it on various levels of details. So Levy type functions are the functions where mass can survive the push forward without consolation only on Levy disk. So being a bit more precise, if the mass loss in push forward of q by a Levy type function and g is small, then almost all mass of q is located on Levy disks for g of big modulus. And big modulus is important. So now to the precise definition of Levy type function, let p be a finite set. And again, you may think about the singular set and g be an entire function. Then we say that g is Levy type. If for any quadratic differential, q with poles in p, any modulus m, any fraction of mass outside disk epsilon, there is a fraction of preserved mass eta such that once the mass constellation is small, so the fraction of preserved mass is bigger than eta. Then there are m Levy disks for function g of modulus bigger or equal than m such that all but epsilon mass sits on these disks. And this definition might sound a bit technical, but in fact, in the exponential paper, it was proven that exponential family of functions are Levy type functions. And another result that we have proven recently is that these multi-era functions are also Levy type functions. And if you have a look at the formula, it doesn't tell you too much. But in fact, functions of this kind, where p is a polynomial, are the functions with finally many acycotic tracks, and that's why they have interest to us and they really have some meaning. And why Levy type functions are so nice is because of the Levy disk principle. And Levy disk principle tells us, if you have some topological post-singular function f, then you assume that all of the entire functions in its parameter space are Levy type. Then such topological model is realized unless there is a Levy cycle. So Levy abstractions are only possible abstractions for function f, for which all the entire functions are Levy type. And here I have to apologize because it's the only slide in my presentation that requires some detailed knowledge of person iteration. But I want to say a couple of words about the proof, about the ideas of the proof of Levy disk principle. All right, so assume that our function is not realizable. And this necessarily means that there is an efficient, so-called efficient sequence of functions gn and quadratic differential qn along the orbit of our iteration in tight mirror space. And efficient means that the ratio of mass cancellation, well, this ratio goes to 1. And functions gn are Levy type, and this means that far enough along the orbit almost all mass sits on Levy disks, like you can see here on the picture. And I hope people online can switch from the slides to the blackboard to see what I have drawn here. You hear a bunch of Levy disks for function gn, and we know that almost all mass sits on them, and they have big modulus. And we can map them forward by function gn into some other moduli, any line of big moduli. And when we do this sufficiently many times, we map them around, and it necessarily has to happen for some Levy disks like this, that in a finite number of iterations that depends only on combinatorics of the function, it travels around, and its image comes back into its homotopy class. OK, so this means that the core curve of this annulus has the cyclic behavior, and if you remember the definition of a Levy disk, the biomorphicity of the map on it was a key point in the definition. So this means that all of the functions between core curves will be one to one, and this red curves on the picture give you a Levy cycle. So this is the proof, if the function is not realizable, then you get your Levy cycle. OK, so the... So you're claiming here a theorem which 50 people have worked on 20 years, and I feel you were a little swift. OK. I didn't actually manage to understand your definition of a Levy disk. I'd like you to go back to it. I mean, if you could actually justify that that's theorem is true, well, you have a fields metal coming to you. That's an interesting statement. I mean, it's not something to dismiss by just one slide. Well... I would really love to go into the details of the proof and explain them to you. Unfortunately, I don't think that I have time for this right now, and also I'm pretty sure that some of the people in the audience won't be able to really follow the discussion, but I definitely would be happy to convince you that this principle, which I don't claim to be proven at the moment, that this principle is true and I'm really going with the fields metal. OK. Yeah, but let me maybe quickly... Part one, we've all known... Yes. Well, my claim would be that the Levy disk principle is somehow... You can already read this through the lines in the exponential paper at this moment. So I don't think that this is such a groundbreaking thing, but again, let's maybe discuss it a bit afterwards, because if the Levy principle is a dynamic type statement, then the following theorem that actually came to be a theorem is a functional statement. It does not involve any dynamics, and it tells you if F and G are two Levy type functions, then the composition is also Levy type. Well, and we're kind of... I gather that the condition that what makes this not absolutely revolutionary is the hypothesis that all the functions G and Levy type. Right. Proving that all the functions of Levy type is really all the work that has to be done in order to put this to practice. Okay, so that's really the hypothesis that makes this theorem. Right. Okay. Okay. Yeah, so let us get back to our functional statement. And I was saying that we already kind of used this theorem as an idea for the Levy disk principle. So assume that the mass constellation for the composition is small, and then the mass constellation for each of the functions must be small due to the properties of push forwards that I mentioned earlier. So you take the Levy disks, you take the Levy disks for function G, and you map them forward, and the disks, well, kind of you can see it on this picture, the images of the disks will be almost the Levy disks for the next function F, just because they share almost all mass of correct differential Q. And then you have actually to think about a bit about the geometry of what's going on, and you have to tweak the mass constellation ratio. But in the end, you can get arbitrary big modulus and arbitrarily small fraction of mass outside of the Levy disk for the intersection of these images of Levy disks for G and Levy disks for F. So this is also, I wouldn't call this theorem a ground breaking result. But what do the Levy disk principle and the Levy composition, the Levy disk composition theorem seem to imply is that if you have two functional families for which, for each of them, Levy abstraction are the only possible type of abstraction, then for the composition, also Levy abstraction should be only possible type of abstraction, and your topological model should be realizable otherwise. So where do we go from here? One goal is to put more functions in the Levy type. This is perhaps what John would be excited to hear about. And I also work on this in my PGD thesis. So this will be in the thesis as well. There is some more work coming this direction. So the next thing, you can actually add critical points to the framework of Levy disks. If you remember, Levy disks mapping one to one was an important piece of definition, but you can actually relax it. You can allow for branched mappings, and you can develop the theory of so-called branch disks and branch disk principle. And for example, this helped me to get a proof for two-genometric family, which as you can see is a composition of a function with two critical points with exponential Levy type function. And maybe a small remark here. You see that the rational function has a pole, but the composition is entire. And I work with entire functions. It would be sufficient for me to have an entire composition. Well, in the final, and perhaps the most ambitious goal for me is to add the poles into the framework of Levy disks. So this would require quite some work because we should see all the other possible obstructions, not only Levy cycles, but all the other obstructions. They come with poles. And this would require understanding the paper, the rational paper, by Duodin Hubbard well enough. So this is a challenging task. But at this point, we believe that this should be doable. So I hope I told you some interesting stuff about transcendental Thurston theory and even for people who are not really aware about Thurston theory too much before this conference. And thank you for your attention.
Thurston theory is a part of my PhD thesis and still work in progress. The history of Thurston geometry. Well, but I work on transcendental at this point that all the functions that I will infinite-dimensional Thurston theory and topological model, have be realized by some If we have a topological post-singular finite function f, and then it's Thurston equivalent to represents a point in Taichmaner space. This is an invariant topological multi-curve with a point of my talk. The first definition that I parameter space are Levy type. Then such topological model is realized unless there is a possible abstractions for function f, for which