Near-degenerate regime in neutral renormalization
Dudko, Dzmitry
A fundamental fact about Riemann surfaces is that they degenerate in a specific pattern — along thin annuli or wide rectangles. As it was demonstrated by W. Thurston in his realization theorem, we can often understand the dynamical system by establishing a priori bounds (a non-escaping property) in the near-degenerate regime. Similar ideas were proven to be successful in the Renormalization Theory of the Mandelbrot set. We will start the talk by discussing a dictionary between Thurston and Renormalization theories. Then we will proceed to neutral renormalization associated with the main cardioid of the Mandelbrot set. We will show how the Transcendental Dynamics naturally appear on the renormalization unstable manifolds. In conclusion, we will describe uniform a priori bounds for neutral renormalization — joint work with Misha Lyubich.
 So we consider the space of all quadratic like maps up to a fine conjugacy. This is a nice analytic space. The renormalization operator would be iteration and restriction. Both are very analytic operations. So the renormalization operator would itself be analytic and it will be contracting. So if we have two maps that are hybrid conjugate, then the renormalizations will also be hybrid conjugate and the quasi conformal distance will be smaller. A caveat is that the dimension of this space is infinite. Well, and our goal is again to find a fixed point or more generally renormalization horseshoe. And for that, we do want some compactness property because since our space is infinite dimension, we can't go far without any compactness property. And it is often called a priori bounds in this subject. And what happens is that often a priori bounds can be established in the near degenerate regime. And this is the main topic of my subject. I think I have some echo. Okay, not anymore. I'd like first to summarize what is known about the near degenerate regime for the first time and quadratic like renormalization theory. And then I'll move to the neutral renormalization. A fundamental fact about the remand surfaces is that they degenerate along thin analyte and white rectangles. A more familiar case is the situation with the compact remand surfaces, perhaps punctured by many points. So in this case, the surface degenerates along thin analyte. So it is called the syncyc decomposition. And if we have on the other hand a remand surface with boundary, so let's assume that we have some Jordan domain in the complex plane minus finitely many Jordan domains. Then such object degenerates along finitely many white rectangles. So it is more convenient to draw not the realistic picture, but a schematic one. So let's apply Photoshop and replace all white rectangles with narrow rectangles. So the convention from now on would be that narrow on pictures means white in reality. So this is because we are interested in homotopical properties of white rectangles. In the Boston theory, when we iterate, if under the iteration we develop long analyte, then there is a certain matrix that controls the degeneration. First of all, if the degeneration develops, then it develops along invariant, multi-curve, and applying the Grotchen equality, we easily obtain an estimate from below of the moduli of the analyte of the pullback through the certain matrix and the moduli of the surface before the pullback. And in fact, in the degenerate situation, this inequality can be effectively reversed and replaced by equality perhaps with an additive correction. Well, this is because, well, our map on analyte is a covering map, and we understand what's happening with covering maps. And the map on thick parts would be belong to some compact set. And this is the idea of how we establish an invariant compact set under the certain iteration. So of course, I skipped a lot of details. In the quadratic life case, so I'm going to consider the primitive bounded case, and I will discuss the argument by Jeremy Khan. So let's consider some quadratic polynomial and just assume that we are in the degenerate situation, so the modulus between u and j is very small. So let's apply renormalization and replace the Julia set by a cycle of small Julia sets. And our task, our goal is to understand what are the white rectangles between small Julia set and the external boundary. And then try to catch the contradiction. And as we can see on the picture, there are two types of white rectangles, horizontal, that connect two small Julia sets, and vertical. And remember that in the certain case, the generation develops along an invariant multi-curve. Well, same story here, that horizontal rectangles must be developed along an invariant graph that connects small Julia sets. But it's full clear that the only invariant graph that connects small Julia set is the hybrid tree. So therefore, horizontal rectangles must be developed along the hybrid tree, but we can say even more. So if we consider the associated PCF model, or it has the hybrid tree, so we can look at the corresponding matrix. Well, and just like the certain matrix controls what is happening with long annulite in the pullback iteration. This matrix controls what's happening with the horizontal rectangles. And in particular, if we apply a grudge type inequality, and if we assume that the core entropy of F restricted to the hybrid tree is positive, then it would tell us that the horizontal rectangles are not very good. So if we assume that there are no vertical rectangles, then this would already lead to a contradiction, well, more or less the same argument as in the Kyrsten case. But a correct statement is that vertical vertical rectangles must form a substantial part of all vertical rectangles. So horizontal rectangles cannot dominate. This is a key step in the Jeremy's argument for the MLC for bounded primitive parameters. Dima, you are proving something. What are you trying to prove? What is the statement? Well, I'm trying to make an analogy between the quadratic life renormalization and Kyrsten. You're making an argument that something is a contradiction of something. What is the statement you're trying to prove? Well, the statement is that the vertical rectangles form a substantial part of all white rectangles. So if we take, so that they will exist at constant C, so that if we sum up, yes. What are these rectangles? Well, rectangles, these are rectangles that are in the same, in the composition of the Riemann surface. When we take you and she removed for the cycle of small Julia sets. So this is a Riemann surface with a boundary. So this surface has the same thing, the composition. Well, elements of the same thing, the compositions will be white rectangles. And the statement is that, well, there was a true statement is that this white rectangle that are horizontal are aligned with the cupboard tree. And the second statement is that if we sum up the width of vertical rectangles, then this would form a substantial part of all rectangles. So I'm trying to, you said we have to Photoshop the picture you're drawing. And I'm trying to do this in my brain. Let me see whether I got it right. The blue rectangles, contrary to what you've drawn, mean that this little Julia sets are much closer to each other than kind of the diameter. So this is the realistic picture. Well, this is the other kind, but the blue ones, the horizontal ones. So the idea is that the little Julia sets are closer to each other than their diameter by some order of magnitude. And so this is two boundary components. And the other boundary components are from the little Julia sets to the outer ellipse boundary, the boundary of you. And here you're saying again that the little Julia sets are closer to the boundary of you than their diameter. Yeah, you can say it in this way. And I'm trying to Photoshop my mental picture here. And then you're saying of all the area in you that we see minus the little Julia set, this is essentially filled by your rectangles. And you're saying you want to say that the definite fraction is of these rectangles is between the little Julia sets and the boundary as opposed to the blue ones between the Julia sets. Correct. Yeah. So this is the statement that Hemel wanted. I ought to be able to understand what it means. I don't know why things are being called rectangles. And I've never seen this language before. And I just don't know what it is that's being talked about. Horizontal sides on these disks or on the boundary of the big D. Assume that you have a disk with several holes, just a few holes. Your rectangle is a topological rectangle whose horizontal sides on the boundary of this Riemann surface. And then you can, so it is just a topological rectangle whose horizontal sides are on the boundary of our Riemann surface in topological and on trivial guys. And the statement is the degeneration of such Riemann surfaces happens when this rectangle, some of these rectangles become white. And so that is the issue of renormalization theory as Dima was trying to convey is to prevent this degeneration, this kind of degeneration. And it is in a way very much similar to how Sursan prevents degeneration along long tubes in the case of compact Riemann surface of finite type. So, okay. Well, in fact, this is what I wanted to say about the quadratic like renormalization. So namely, I wanted to make this parallel that if you look at what's happening, then the argument is indeed kind of very close to what's happening in the Sursan iteration. And that's kind of one of the key steps, if not the key step in Jeremy Khan proof of the MLC in the primitive bounded case. An important tool is the covering lemma developed by Jeremy and Misha. And in the follow up work, Misha and Jeremy push the theory much farther. So now it handles quite well the primitive case, assuming that we are kind of uniformly away from the molecule. So roughly speaking, assuming that the core entropy is at least some epsilon. So roughly what they did, they translated all the theory of the primitive renormalization into the language of the near degenerate regime. Yeah, and establish a priori bounds under such kind of condition. Okay, so more details will be tomorrow in Misha's talk. So this is just before the bulliness. So please come to bulliness. So let me also make a comment about the Sursan theory. So there is another approach to the renormalization to the realization problem by Dylan. So he was talking about it yesterday. And in some sense, the near degenerate regime is even more explicit in his situation. So he starts with a map between graphs. So then he replaces graphs with a small neighborhood. This will be a very degenerate remand surface, and then he does the realization for this degenerate remand surfaces. And then after that, he does the surgery and obtains epispirically finite rational map. So Kevin H. Dylan will be given a Zoom mini course in November on the renormalization and dynamics seminar. So everyone who is interested to the subject is welcome to join. Okay, good. So much now. I'd like to go to the near neutral renormalization theory. So this is the theory associated with the boundary of the main capoboli component. So here are some examples of maps. So all of them are digital disks, but some of these digital disks are degenerate. So this is the golden digital disk. So this digital disk is a quasi disk that rotates around the center. So this digital disk has the rotation number that is very close to zero. So it is still a quasi disk, but the center of rotation will be very close to the boundary. So here, the rotation number is close to one half. And then the digital disk will be like this. So it will start to develop parabolic fjords. But still it's kind of a nice object. So as I will talk later. So here the rotation number is close to one quarter. And we can see that the digital disk has four parabolic fjords. So it rotates. So it's not, well, it wants to degenerate and it develops parabolic fjords. So there are several reasons, or there are many reasons why we might be interested in the neutral renormalization. First, the neutral renormalization controls the quadratic like renormalization near the cardioid. So this is where the primitive renormalization theory fails. It requires certain adjustments. In particular, there is no good theory of puzzles near the boundary of the cardioid. Well, also, near neutral renormalization is very much related to the theory of near-circle maps and such maps were in the focus for a long time. Well, perhaps even from the beginning of the dynamical systems, when the dynamical systems were motivated by the celestial mechanics like planets moving around the sun. Well, and since we like the transcendental dynamics, at least on this conference, well, a good news is that the transcendental dynamics also emerges on the unstable manifolds. So this is kind of one way how we can understand combinatorics of neutral renormalization. Namely the combinatorics can be described using this particular map. So this is the Julia set of this cubic polynomial. So it has a parabolic point here, and it attracts one critical point, and another critical point is here, and it maps to the fixed point under one iteration, S221. So this field, Julia set, is very similar to the molecule, and in fact, if we endow the molecule with the Brunner DOD surgery, and I will refer to the resulting map as the molecule map, then this should be conjugate to this topologically. Well, in fact, this is equivalent to the MLC conjecture, the satellite case, that these two dynamical systems are topologically conjugate. And the Brunner DOD surgery, well, let me kind of describe the idea on the case with rabbits. So if we have a peak of a Q-rabbit, so it's the rabbit that rotates with this rotational number, then assuming that P is smaller than 2 over 2, we can remove dynamically PES from the rabbit. So we do the surgery. Well, and then after the DOD Brunner surgery, we obtain another rabbit, which will have the rotation number P over Q minus P. Well, if P is bigger than Q over 2, then while we symmetrized the construction, well, we also map the basillic apabolic component into the main apabolic component using the basillic straightening map. So everything assembles into a continuous map around the main molecule, and the MLC conjecture of the satellite case say that this must be topologically conjugate to this molecule. And periodic points on the main apabolic component of the molecule map are exactly the legal maps of periodic type. Alternatively, we can describe the legal maps of periodic type using continuous fraction expansion. So this is the periodic point of the smallest period, so here is period 2 of the molecule map. So this periodic cycle is formed by the golden-ziggle parameters, so the upper golden-ziggle parameters and the lower ziggle parameter. Once we have a periodic point, we should expect self-similarity because around periodic points we may want to linearize everything. And this is indeed what happens with the Mandelbrot set. So if we zoom in at the golden-ziggle parameter, then we see self-similarity. So this zoom is almost indistinguishable from the zoom next level. So this is called universality. The combinatorial reason is because the golden-ziggle parameter is a periodic cycle. But of course, in order to justify this self-similarity properly, we need to develop the randomization theory. Let me also make a remark that if we consider some other parameter space, for example, the space of quadratic rational maps, where one of the critical points is the periodic, and if we zoom in at the corresponding golden-ziggle parameter, then again this picture becomes almost indistinguishable from the corresponding picture of the Mandelbrot set. And then the scaling will be with the same universal constant lambda star. So here would also be. And in fact, both of these pictures converge to the limiting transcendental dynamics. And the limiting transcendental dynamics can be used in order to identify these pictures well at least partially, but hopefully in the future there would be a complete identification. Okay, so this is a summary of the neutral randomization theory. So the first conjectures appeared in the 80s. So that was the time when many physicists discovered many interesting facts. So mathematicians call them conjectures. Kurt McMullen constructed randomization ziggle periodic points associated with ziggle disks of a periodic type. He also showed the exponential contraction. So Ina and Shishikura proved the publicity of the neoparabolic randomization. So this randomization controls what happens in the neighborhood of a parabolic point. And this was important for the theory because the publicity really allows to study nearby maps. And what they did, they proved the publicity for the actual parabolic map. And since the publicity is an open condition, it also holds in a neighborhood and it allows to study maps that were out of reach before. For example, there are many applications to the area problem. There is now kind of almost a full or complete topological understanding of the hedgehog of a Kramer parameter in the Ina Shishikura class. So Kramer polynomials turns out to be not that complicated after all. Well, there are applications to the MLC conjecture in the satellite case and also ergodic properties of neutral polynomials. So the original argument by Ina and Shishikura involved some computer assistants are not later presented a computer free proof and she also works for unicritical maps also high type. Denise Guidershuk and Michael Jampolsky presented a computer assistant proof for the actual golden mean, the digital disk, namely for this main periodic cycle. Well, then in a joint work with Mishan Nikita, we proved the publicity for bounded type parameters while we involved the packman renormalization. Well, again, the publicity means that we can study nearby maps. So we understand quite well what is happening with periodic, with digital disk or periodic type. And once we have publicity, we can study also nearby maps, and we obtain some applications to also scaling and area problems. And we constructed the first example of the MLC of the satellite bounded type. Let me also mention briefly the Geodigis surgery. So this is a way how we understand digital disks of bounded type. Otherwise, we can't control the critical orbit. So the idea is that we start with the Blaschke product, a Blaschke product restricts to a circle map. So it's going to be a critical circle map. And since the critical circle maps are quite well understood, in particular, if our rotation number is of bounded type, then the restrictions to the circle will be QC conjugate to the rotation. Then this allows to apply a quasi conformal surgery and obtain out of the Blaschke product a Ziegler polynomial of periodic type. So we forget the dynamics inside the unit disk and we do the surgery, and this is an essential bit for the surgery. And what's important is that we obtain a map where we control the past critical orbit. The past critical orbit would be rotated on the boundary of the Ziegler quasi disk. But there are limitations for this surgery because the surgery cannot produce a framework polynomials. Okay, so let me now state Ethereum, so it's a joint work with Misha. So the theorem states that there are uniform a priori bounds for all neutral polynomials on the boundary of the capoboli component. So there are different ways of stating it. So let me first state kind of a rough version. So a rough version claims that when the Ziegler disks of a bounded type degenerate, then they can degenerate only in a star-like way. So as we can see on the picture, so the degenerations has this type, but we don't see the degenerations like this. So this is we don't see on pictures. And this is the statement, so it can be made precise, but roughly speaking, when a Ziegler disk degenerate, then this is allowed. So such degenerations is allowed, and it can also happen on all scales. So this is like scale, let's say one, and if we zoom in, then we would see more parabolic fjords, and still there would be star-like patient. And this is not allowed. So a bit more precise way of stating it is that for any rotation number of a bounded type, but M can be any Ziegler disk, so this is our Ziegler disk that rotates. This Ziegler disk can be put inside a quasi-disk with a uniform dilatation, such that the degree of the map restricted to this enlarged quasi-disk, so we call it pseudo-ziegler disk. So such that the degree is one of the map restricted to this pseudo-ziegler disk. And as a corollary, because quasi-conformal objects are compact, such pseudo-ziegler disk exists for all neutral maps, even for Kramer polynomials, so there would be a pseudo-ziegler disk that is sort of almost invariant, and this pseudo-ziegler disk would have an absolute dilatation. So these pseudo-ziegler disks are obtained by adding these parabolic fjords on all scales. So this is how we construct the Ziegler disk. Dima, it's not really visible. So Raph is speaking out of a star. We can produce a... Dima, could you... Accuracy. Dima, could you please repeat your drawing, because it was not very visible. It's all dark. Is it better now? Are you talking about this? Yes. Maybe you can make it closer to us. Right. So is it better? Yes. Is it? Right. So what I'm saying is that pseudo-ziegler disk can be obtained by adding such parabolic fjords, while in stimuli with other periodic Ziegler parameters. So whenever a parabolic fjords is developed, we can add it to the object. So I was saying it more schematically. If we have a star, then we can add fjords between star and obtain a uniform QC disk. Right. And another way of stating a priori bounce is that there would exist a fiber-wise compact renomization operator, cylinder renomization operator around MacMallon periodic points when we take the closure of MacMallon periodic points. So this is renomization horseshoe. So we take all MacMallon periodic renomization points. We take the closure and the statement is that this will be a compact set of maps and there would be a renomization operator around this compact set of maps. So any questions about the statement? So any questions at this stage? Before I'll try to give some ideas of the proofs. Okay, good. So as I said, the proof is in the near-degenerate regime and here is like one basic idea that suppose we have a single disk that rotates. And suppose we have two intervals, i and j. And suppose that there is a vitric tunnel that connects them. So again, this is kind of a photoshop, but in reality this vitric tunnel will be like this. Well, as now, if combinatorially we can pull back these two intervals, i and j, two intervals located like this, so between them. And if we assume that the critical value is not on the way, then we can pull back this vitric tunnel and obtain a vitric tunnel that cross intersects the original vitric tunnel. And this is impossible because vitric tunnels do not cross intersect. And this type of arguments allows to cover like a lot of cases. So a good half of all the arguments are based on this principle. So in particular, this type of arguments allows to understand the geometry of parabolic fjords. Let me now kind of try to give a cartoon proof of the logic. So this will be like a cartoon rough presentation of the main logic of the argument. Assume that we don't have uniform a priori bounds. So this means that we have a digital disk that looks like this in reality. So like one peninsula approach becomes very close to another peninsula. Then there is a vitric tunnel between these two peninsulas. So as I said, we prefer to do schematic pictures. So if we apply the Photoshop, then we obtain like we put our stuff back, but then we would have a vitric tunnel between these two peninsula. So vitric tunnels geometrically looks like this. Well, it's now this degeneration can be spread around. So the digital disk rotates. The covering lemma allows to kind of spread this degeneration around to more or less all the peninsula, not more or less, but to all peninsulas. But now if we try to spread this rectangle to a rectangle that connects these two peninsula, then this is impossible because vitric tunnels cannot cross intersect. So therefore, when we apply the covering lemma and move the original rectangle to the rectangle that connects this two peninsula, the new rectangle must go through the digital disk. So it must be submerged. Well, and this submergence could be understood because inside the unit disk we have the complex structure of the standard disk that rotates. So in particular, these degeneration can be localized. So more of the degeneration would go like this. And like only a big of one of the degeneration would be beyond, well, would be below this. We call these techniques like snakes. And we have several versions of the snake lemma. Right. But now after localizing the degeneration, we can also observe that roughly this degeneration fits into two rectangles. And one of these rectangles should be twice wider than the original rectangle. So this allows kind of us to say that either this rectangle or this rectangle will be twice bigger than the original rectangle. So we can obtain from a degeneration, bigger degeneration. Well, and then this bigger degeneration could be put inside a fjord of a, inside the epininsular of a deeper scale. But here comes sort of the main logic that if we originally had a big degeneration, then we can find bigger and bigger degeneration at its depth and we can construct like a sequence of rectangles so that each rectangle I n will be to the power n wider than the original rectangle. But such sequence cannot exist because, because we do have a non uniform a priori bounds. So our digital disk was of periodic type. So it might be kind of super degenerate, but still it is of periodic type and it has some a priori bounds. So therefore, therefore, this is impossible and it implies that we cannot find such degenerations at first place. And that's kind of a rough kind of cartoon idea of the argument that allows to promote non uniform bounds to uniform bounds. So let me make like a few final comments like first, as I said, we need to work with the digital disk, so we add parabolic fjords to the digital disk. And we do make such construction on all scales, so to catch part of the technicality is sort of to show that different levels don't really interact much. So this is the way how we do it is by establishing a certain force a priori bounds. Well, another remark that kind of maybe let me say like about applications. So since this should be digital disk kind of exists for all data, so this would imply that for every data there would be some HCOG, let's say for every data that is not rational for every rational data, there will be a HCOG that contains the critical orbit. And for that, that agree is one. So we can't kind of claim at this stage the topology of that patch call, but at least it would exist and the degree will be one. So in particular, the post-critical set is not everything, which is an open question outside of the Inno-Shishikura class. But in the Inno-Shishikura class, so as I said, there is now full topological understanding of such objects and she are hopefully in kind of in the future, this topological understanding can be improved for all neutral polynomials. Let me stop here. Okay. Thank you. Are there any questions? You are online? Okay. Maybe I have a question. So Jeremy's philosophy was something like if your life is bad today, then your life was even worse yesterday. And your philosophy is that if your life is bad today, then it will be even worse tomorrow. So this is a joke, but do you have a dynamical or more like serious explanation why it goes another way? I mean, the reason is this, Naik, Lema. Well, in the Jeremy situation, let me go back to the Jeremy situation. Well, the Julia sets are disjoint and he doesn't need to kind of understand kind of the boundary of each, the geometry near the boundary of each small Julia set. He can kind of think that each small Julia set is roughly kind of a disk for this type of arguments. So here, here the Ziggle disk is one object, so it is a connected object. So we need to kind of understand the ideal boundary of the Ziggle disk with respect to the geometry outside of the Ziggle disk. So this kind of leads up to understanding the submergence of the white rectangles inside the Ziggle disk. And this sort of forces this philosophy that this kind of forces to go into the deep scale instead of going into the shallow case, into this shallow scale. So this is really because we need to understand the submergence of white rectangles inside the Ziggle disk. Yeah, let me also mention that I gave a mini course, a full lecture mini course last December about more details of this type of argument. So it's available on the Simon Center video collection. More questions? Can I ask a question? Sure. Yeah. So I'm wondering if you are also able to control the geometry of the fjords within this pseudo Ziggle disk that you talked about. For example, the fjords start to say a spoiler on the inside. A spoiler about the fixed point or to do other things. Right. Yeah, we don't. Right. Yeah. So in some sense, our arguments allow us to understand the external boundary. So our arguments sort of allow us to understand the external boundary of this pseudo Ziggle disk. But we can't really say much, I guess, at this stage about what's happening around the fixed point. So we don't. Yeah, we can't say how fjords would spiral. Right. So I mean, maybe a different question is if I understand correctly, the pseudo Ziggle disk contains, say, the first few iterates of the critical point. Yeah. Is this right? Maybe could I really state your theorem that for any, like, no finite number m, there exists a k such that for any bound, state of bounded type, there is this pseudo Ziggle disk, which contains the first m iterates of the critical point. Yes, this is correct. So we may need to kind of enlarge k, the capital K, but yeah, by enlarging this absolute k, we can control much more pseudo Ziggle disks. We can control much more the perspective orbit. Yes. Right. Because without stating it like that, because it seems to me that there could be maybe a trivial example of a pseudo Ziggle disk, you just take a closed curve which passes through the critical point. Yeah. And the map restricts to a degree warm up. No, but. Because your pseudo Ziggle disk is not invariant. Yeah, but pseudo Ziggle disks does contain the original Ziggle disk. And it is kind of an open question. Well, I mean, it was an open question kind of why the boundary of the Ziggle disk is not everything. I have a question about higher degree polynomials. So for instance, in the family of all degree three polynomials, would you also have a uniform k such that your theorem holds? Right. I think, I mean, kind of what we really do. So we start with the Ziggle disks that we understand with that are non-uniform. And then we kind of provide a priori bounds for them. Well now for Cubics. I mean for Cubics, if we consider a Ziggle disk of periodic type, perhaps a non-uniform that rotates. And if this pseudo Ziggle disk contains both critical points on the boundary, then I would expect that the technique should be applicable perhaps with minor modification. So in particular, such objects kind of are compact or at least I would anticipate it. But in the cubic case, it can may happen that one, I mean, we know that one of the critical points must be on the boundary. This is thanks to Joe Digi's surgery and Schmitz's show how it can be applied in the multi-critical for multi-critical polynomials. So if we have a Ziggle disk of periodic type, then it must contain one of the critical points. But then another critical point must be sort of kind of on the decorations. This may be kind of a challenge. Because I mean, if you want to understand all neutral cubic polynomials, then we want to understand also the parameter that are obtained by taking limits of these objects. Okay, thank you. Okay, let us send the speaker again. Thank you, Jeff. Okay, thank you.
 So we consider the space of all quadratic like maps up to a fine conjugacy.  This is a nice analytic space.  The renormalization operator would be iteration and restriction.  Both are very analytic operations.  So the renormalization operator would itself be analytic and it will be contracting.  So if we have two maps that are hybrid conjugate, then the renormalizations will also be hybrid  conjugate and the quasi conformal distance will be smaller.  A caveat is that the dimension of this space is infinite.  Well, and our goal is again to find a fixed point or more generally renormalization horseshoe.  And for that, we do want some compactness property because since our space is infinite dimension,  we can't go far without any compactness property.  And it is often called a priori bounds in this subject.  And what happens is that often a priori bounds can be established in the near degenerate  regime.  And this is the main topic of my subject.  I think I have some echo.  Okay, not anymore.  I'd like first to summarize what is known about the near degenerate regime for the  first time and quadratic like renormalization theory.  And then I'll move to the neutral renormalization.  A fundamental fact about the remand surfaces is that they degenerate along thin analyte  and white rectangles.  A more familiar case is the situation with the compact remand surfaces, perhaps punctured  by many points.  So in this case, the surface degenerates along thin analyte.  So it is called the syncyc decomposition.  And if we have on the other hand a remand surface with boundary, so let's assume that  we have some Jordan domain in the complex plane minus finitely many Jordan domains.  Then such object degenerates along finitely many white rectangles.  So it is more convenient to draw not the realistic picture, but a schematic one.  So let's apply Photoshop and replace all white rectangles with narrow rectangles.  So the convention from now on would be that narrow on pictures means white in reality.  So this is because we are interested in homotopical properties of white rectangles.  In the Boston theory, when we iterate, if under the iteration we develop long analyte,  then there is a certain matrix that controls the degeneration.  First of all, if the degeneration develops, then it develops along invariant, multi-curve,  and applying the Grotchen equality, we easily obtain an estimate from below of the moduli  of the analyte of the pullback through the certain matrix and the moduli of the surface  before the pullback.  And in fact, in the degenerate situation, this inequality can be effectively reversed  and replaced by equality perhaps with an additive correction.  Well, this is because, well, our map on analyte is a covering map, and we understand what's  happening with covering maps.  And the map on thick parts would be belong to some compact set.  And this is the idea of how we establish an invariant compact set under the certain iteration.  So of course, I skipped a lot of details.  In the quadratic life case, so I'm going to consider the primitive bounded case, and  I will discuss the argument by Jeremy Khan.  So let's consider some quadratic polynomial and just assume that we are in the degenerate  situation, so the modulus between u and j is very small.  So let's apply renormalization and replace the Julia set by a cycle of small Julia sets.  And our task, our goal is to understand what are the white rectangles between small Julia  set and the external boundary.  And then try to catch the contradiction.  And as we can see on the picture, there are two types of white rectangles, horizontal,  that connect two small Julia sets, and vertical.  And remember that in the certain case, the generation develops along an invariant multi-curve.  Well, same story here, that horizontal rectangles must be developed along an invariant graph  that connects small Julia sets.  But it's full clear that the only invariant graph that connects small Julia set is the  hybrid tree.  So therefore, horizontal rectangles must be developed along the hybrid tree, but we can  say even more.  So if we consider the associated PCF model, or it has the hybrid tree, so we can look  at the corresponding matrix.  Well, and just like the certain matrix controls what is happening with long annulite in the  pullback iteration.  This matrix controls what's happening with the horizontal rectangles.  And in particular, if we apply a grudge type inequality, and if we assume that the core  entropy of F restricted to the hybrid tree is positive, then it would tell us that the  horizontal rectangles are not very good.  So if we assume that there are no vertical rectangles, then this would already lead  to a contradiction, well, more or less the same argument as in the Kyrsten case.  But a correct statement is that vertical vertical rectangles must form a substantial part of  all vertical rectangles.  So horizontal rectangles cannot dominate.  This is a key step in the Jeremy's argument for the MLC for bounded primitive parameters.  Dima, you are proving something.  What are you trying to prove?  What is the statement?  Well, I'm trying to make an analogy between the quadratic life renormalization and Kyrsten.  You're making an argument that something is a contradiction of something.  What is the statement you're trying to prove?  Well, the statement is that the vertical rectangles form a substantial part of all white rectangles.  So if we take, so that they will exist at constant C, so that if we sum up, yes.  What are these rectangles?  Well, rectangles, these are rectangles that are in the same, in the composition of the  Riemann surface.  When we take you and she removed for the cycle of small Julia sets.  So this is a Riemann surface with a boundary.  So this surface has the same thing, the composition.  Well, elements of the same thing, the compositions will be white rectangles.  And the statement is that, well, there was a true statement is that this white rectangle  that are horizontal are aligned with the cupboard tree.  And the second statement is that if we sum up the width of vertical rectangles, then  this would form a substantial part of all rectangles.  So I'm trying to, you said we have to Photoshop the picture you're drawing.  And I'm trying to do this in my brain.  Let me see whether I got it right.  The blue rectangles, contrary to what you've drawn, mean that this little Julia sets are  much closer to each other than kind of the diameter.  So this is the realistic picture.  Well, this is the other kind, but the blue ones, the horizontal ones.  So the idea is that the little Julia sets are closer to each other than their diameter  by some order of magnitude.  And so this is two boundary components.  And the other boundary components are from the little Julia sets to the outer ellipse  boundary, the boundary of you.  And here you're saying again that the little Julia sets are closer to the boundary of you  than their diameter.  Yeah, you can say it in this way.  And I'm trying to Photoshop my mental picture here.  And then you're saying of all the area in you that we see minus the little Julia set,  this is essentially filled by your rectangles.  And you're saying you want to say that the definite fraction is of these rectangles is  between the little Julia sets and the boundary as opposed to the blue ones between the Julia  sets.  Correct.  Yeah.  So this is the statement that Hemel wanted.  I ought to be able to understand what it means.  I don't know why things are being called rectangles.  And I've never seen this language before.  And I just don't know what it is that's being talked about.  Horizontal sides on these disks or on the boundary of the big D.  Assume that you have a disk with several holes, just a few holes.  Your rectangle is a topological rectangle whose horizontal sides on the boundary of  this Riemann surface.  And then you can, so it is just a topological rectangle whose horizontal sides are on the  boundary of our Riemann surface in topological and on trivial guys.  And the statement is the degeneration of such Riemann surfaces happens when this rectangle,  some of these rectangles become white.  And so that is the issue of renormalization theory as Dima was trying to convey is to prevent  this degeneration, this kind of degeneration.  And it is in a way very much similar to how Sursan prevents degeneration along long tubes  in the case of compact Riemann surface of finite type.  So, okay.  Well, in fact, this is what I wanted to say about the quadratic like renormalization.  So namely, I wanted to make this parallel that if you look at what's happening, then  the argument is indeed kind of very close to what's happening in the Sursan iteration.  And that's kind of one of the key steps, if not the key step in Jeremy Khan proof of the  MLC in the primitive bounded case.  An important tool is the covering lemma developed by Jeremy and Misha.  And in the follow up work, Misha and Jeremy push the theory much farther.  So now it handles quite well the primitive case, assuming that we are kind of uniformly  away from the molecule.  So roughly speaking, assuming that the core entropy is at least some epsilon.  So roughly what they did, they translated all the theory of the primitive renormalization  into the language of the near degenerate regime.  Yeah, and establish a priori bounds under such kind of condition.  Okay, so more details will be tomorrow in Misha's talk.  So this is just before the bulliness.  So please come to bulliness.  So let me also make a comment about the Sursan theory.  So there is another approach to the renormalization to the realization problem by Dylan.  So he was talking about it yesterday.  And in some sense, the near degenerate regime is even more explicit in his situation.  So he starts with a map between graphs.  So then he replaces graphs with a small neighborhood.  This will be a very degenerate remand surface, and then he does the realization for this  degenerate remand surfaces.  And then after that, he does the surgery and obtains epispirically finite rational map.  So Kevin H. Dylan will be given a Zoom mini course in November on the renormalization  and dynamics seminar.  So everyone who is interested to the subject is welcome to join.  Okay, good.  So much now.  I'd like to go to the near neutral renormalization theory.  So this is the theory associated with the boundary of the main capoboli component.  So here are some examples of maps.  So all of them are digital disks, but some of these digital disks are degenerate.  So this is the golden digital disk.  So this digital disk is a quasi disk that rotates around the center.  So this digital disk has the rotation number that is very close to zero.  So it is still a quasi disk, but the center of rotation will be very close to the boundary.  So here, the rotation number is close to one half.  And then the digital disk will be like this.  So it will start to develop parabolic fjords.  But still it's kind of a nice object.  So as I will talk later.  So here the rotation number is close to one quarter.  And we can see that the digital disk has four parabolic fjords.  So it rotates.  So it's not, well, it wants to degenerate and it develops parabolic fjords.  So there are several reasons, or there are many reasons why we might be interested in  the neutral renormalization.  First, the neutral renormalization controls the quadratic like renormalization near the  cardioid.  So this is where the primitive renormalization theory fails.  It requires certain adjustments.  In particular, there is no good theory of puzzles near the boundary of the cardioid.  Well, also, near neutral renormalization is very much related to the theory of near-circle  maps and such maps were in the focus for a long time.  Well, perhaps even from the beginning of the dynamical systems, when the dynamical systems  were motivated by the celestial mechanics like planets moving around the sun.  Well, and since we like the transcendental dynamics, at least on this conference, well,  a good news is that the transcendental dynamics also emerges on the unstable manifolds.  So this is kind of one way how we can understand combinatorics of neutral renormalization.  Namely the combinatorics can be described using this particular map.  So this is the Julia set of this cubic polynomial.  So it has a parabolic point here, and it attracts one critical point, and another critical point  is here, and it maps to the fixed point under one iteration, S221.  So this field, Julia set, is very similar to the molecule, and in fact, if we endow  the molecule with the Brunner DOD surgery, and I will refer to the resulting map as the  molecule map, then this should be conjugate to this topologically.  Well, in fact, this is equivalent to the MLC conjecture, the satellite case, that these  two dynamical systems are topologically conjugate.  And the Brunner DOD surgery, well, let me kind of describe the idea on the case with  rabbits.  So if we have a peak of a Q-rabbit, so it's the rabbit that rotates with this rotational  number, then assuming that P is smaller than 2 over 2, we can remove dynamically PES from  the rabbit.  So we do the surgery.  Well, and then after the DOD Brunner surgery, we obtain another rabbit, which will have the  rotation number P over Q minus P. Well, if P is bigger than Q over 2, then while we symmetrized  the construction, well, we also map the basillic apabolic component into the main apabolic  component using the basillic straightening map.  So everything assembles into a continuous map around the main molecule, and the MLC conjecture  of the satellite case say that this must be topologically conjugate to this molecule.  And periodic points on the main apabolic component of the molecule map are exactly the legal  maps of periodic type.  Alternatively, we can describe the legal maps of periodic type using continuous fraction  expansion.  So this is the periodic point of the smallest period, so here is period 2 of the molecule  map.  So this periodic cycle is formed by the golden-ziggle parameters, so the upper golden-ziggle  parameters and the lower ziggle parameter.  Once we have a periodic point, we should expect self-similarity because around periodic points  we may want to linearize everything.  And this is indeed what happens with the Mandelbrot set.  So if we zoom in at the golden-ziggle parameter, then we see self-similarity.  So this zoom is almost indistinguishable from the zoom next level.  So this is called universality.  The combinatorial reason is because the golden-ziggle parameter is a periodic cycle.  But of course, in order to justify this self-similarity properly, we need to develop the randomization  theory.  Let me also make a remark that if we consider some other parameter space, for example, the  space of quadratic rational maps, where one of the critical points is the periodic, and  if we zoom in at the corresponding golden-ziggle parameter, then again this picture becomes  almost indistinguishable from the corresponding picture of the Mandelbrot set.  And then the scaling will be with the same universal constant lambda star.  So here would also be.  And in fact, both of these pictures converge to the limiting transcendental dynamics.  And the limiting transcendental dynamics can be used in order to identify these pictures  well at least partially, but hopefully in the future there would be a complete identification.  Okay, so this is a summary of the neutral randomization theory.  So the first conjectures appeared in the 80s.  So that was the time when many physicists discovered many interesting facts.  So mathematicians call them conjectures.  Kurt McMullen constructed randomization ziggle periodic points associated with ziggle disks  of a periodic type.  He also showed the exponential contraction.  So Ina and Shishikura proved the publicity of the neoparabolic randomization.  So this randomization controls what happens in the neighborhood of a parabolic point.  And this was important for the theory because the publicity really allows to study nearby  maps.  And what they did, they proved the publicity for the actual parabolic map.  And since the publicity is an open condition, it also holds in a neighborhood and it allows  to study maps that were out of reach before.  For example, there are many applications to the area problem.  There is now kind of almost a full or complete topological understanding of the hedgehog of  a Kramer parameter in the Ina Shishikura class.  So Kramer polynomials turns out to be not that complicated after all.  Well, there are applications to the MLC conjecture in the satellite case and also ergodic properties  of neutral polynomials.  So the original argument by Ina and Shishikura involved some computer assistants are not  later presented a computer free proof and she also works for unicritical maps also high  type.  Denise Guidershuk and Michael Jampolsky presented a computer assistant proof for the actual  golden mean, the digital disk, namely for this main periodic cycle.  Well, then in a joint work with Mishan Nikita, we proved the publicity for bounded type parameters  while we involved the packman renormalization.  Well, again, the publicity means that we can study nearby maps.  So we understand quite well what is happening with periodic, with digital disk or periodic  type.  And once we have publicity, we can study also nearby maps, and we obtain some applications  to also scaling and area problems.  And we constructed the first example of the MLC of the satellite bounded type.  Let me also mention briefly the Geodigis surgery.  So this is a way how we understand digital disks of bounded type.  Otherwise, we can't control the critical orbit.  So the idea is that we start with the Blaschke product, a Blaschke product restricts to a  circle map.  So it's going to be a critical circle map.  And since the critical circle maps are quite well understood, in particular, if our rotation  number is of bounded type, then the restrictions to the circle will be QC conjugate to the rotation.  Then this allows to apply a quasi conformal surgery and obtain out of the Blaschke product  a Ziegler polynomial of periodic type.  So we forget the dynamics inside the unit disk and we do the surgery, and this is an  essential bit for the surgery.  And what's important is that we obtain a map where we control the past critical orbit.  The past critical orbit would be rotated on the boundary of the Ziegler quasi disk.  But there are limitations for this surgery because the surgery cannot produce a framework  polynomials.  Okay, so let me now state Ethereum, so it's a joint work with Misha.  So the theorem states that there are uniform a priori bounds for all neutral polynomials  on the boundary of the capoboli component.  So there are different ways of stating it.  So let me first state kind of a rough version.  So a rough version claims that when the Ziegler disks of a bounded type degenerate, then they  can degenerate only in a star-like way.  So as we can see on the picture, so the degenerations has this type, but we don't see the degenerations  like this.  So this is we don't see on pictures.  And this is the statement, so it can be made precise, but roughly speaking, when a Ziegler  disk degenerate, then this is allowed.  So such degenerations is allowed, and it can also happen on all scales.  So this is like scale, let's say one, and if we zoom in, then we would see more parabolic  fjords, and still there would be star-like patient.  And this is not allowed.  So a bit more precise way of stating it is that for any rotation number of a bounded type,  but M can be any Ziegler disk, so this is our Ziegler disk that rotates.  This Ziegler disk can be put inside a quasi-disk with a uniform dilatation, such that the degree  of the map restricted to this enlarged quasi-disk, so we call it pseudo-ziegler disk.  So such that the degree is one of the map restricted to this pseudo-ziegler disk.  And as a corollary, because quasi-conformal objects are compact, such pseudo-ziegler disk  exists for all neutral maps, even for Kramer polynomials, so there would be a pseudo-ziegler  disk that is sort of almost invariant, and this pseudo-ziegler disk would have an absolute  dilatation.  So these pseudo-ziegler disks are obtained by adding these parabolic fjords on all scales.  So this is how we construct the Ziegler disk.  Dima, it's not really visible.  So Raph is speaking out of a star.  We can produce a...  Dima, could you...  Accuracy.  Dima, could you please repeat your drawing, because it was not very visible.  It's all dark.  Is it better now?  Are you talking about this?  Yes.  Maybe you can make it closer to us.  Right.  So is it better?  Yes.  Is it?  Right.  So what I'm saying is that pseudo-ziegler disk can be obtained by adding such parabolic  fjords, while in stimuli with other periodic Ziegler parameters.  So whenever a parabolic fjords is developed, we can add it to the object.  So I was saying it more schematically.  If we have a star, then we can add fjords between star and obtain a uniform QC disk.  Right.  And another way of stating a priori bounce is that there would exist a fiber-wise compact  renomization operator, cylinder renomization operator around MacMallon periodic points  when we take the closure of MacMallon periodic points.  So this is renomization horseshoe.  So we take all MacMallon periodic renomization points.  We take the closure and the statement is that this will be a compact set of maps and there  would be a renomization operator around this compact set of maps.  So any questions about the statement?  So any questions at this stage?  Before I'll try to give some ideas of the proofs.  Okay, good.  So as I said, the proof is in the near-degenerate regime and here is like one basic idea that  suppose we have a single disk that rotates.  And suppose we have two intervals, i and j.  And suppose that there is a vitric tunnel that connects them.  So again, this is kind of a photoshop, but in reality this vitric tunnel will be like  this.  Well, as now, if combinatorially we can pull back these two intervals, i and j, two intervals  located like this, so between them.  And if we assume that the critical value is not on the way, then we can pull back this  vitric tunnel and obtain a vitric tunnel that cross intersects the original vitric tunnel.  And this is impossible because vitric tunnels do not cross intersect.  And this type of arguments allows to cover like a lot of cases.  So a good half of all the arguments are based on this principle.  So in particular, this type of arguments allows to understand the geometry of parabolic fjords.  Let me now kind of try to give a cartoon proof of the logic.  So this will be like a cartoon rough presentation of the main logic of the argument.  Assume that we don't have uniform a priori bounds.  So this means that we have a digital disk that looks like this in reality.  So like one peninsula approach becomes very close to another peninsula.  Then there is a vitric tunnel between these two peninsulas.  So as I said, we prefer to do schematic pictures.  So if we apply the Photoshop, then we obtain like we put our stuff back, but then we would  have a vitric tunnel between these two peninsula.  So vitric tunnels geometrically looks like this.  Well, it's now this degeneration can be spread around.  So the digital disk rotates.  The covering lemma allows to kind of spread this degeneration around to more or less all  the peninsula, not more or less, but to all peninsulas.  But now if we try to spread this rectangle to a rectangle that connects these two peninsula,  then this is impossible because vitric tunnels cannot cross intersect.  So therefore, when we apply the covering lemma and move the original rectangle to the rectangle  that connects this two peninsula, the new rectangle must go through the digital disk.  So it must be submerged.  Well, and this submergence could be understood because inside the unit disk we have the complex  structure of the standard disk that rotates.  So in particular, these degeneration can be localized.  So more of the degeneration would go like this.  And like only a big of one of the degeneration would be beyond, well, would be below this.  We call these techniques like snakes.  And we have several versions of the snake lemma.  Right.  But now after localizing the degeneration, we can also observe that roughly this degeneration  fits into two rectangles.  And one of these rectangles should be twice wider than the original rectangle.  So this allows kind of us to say that either this rectangle or this rectangle will be twice  bigger than the original rectangle.  So we can obtain from a degeneration, bigger degeneration.  Well, and then this bigger degeneration could be put inside a fjord of a, inside the epininsular  of a deeper scale.  But here comes sort of the main logic that if we originally had a big degeneration, then  we can find bigger and bigger degeneration at its depth and we can construct like a sequence  of rectangles so that each rectangle I n will be to the power n wider than the original  rectangle.  But such sequence cannot exist because, because we do have a non uniform a priori bounds.  So our digital disk was of periodic type.  So it might be kind of super degenerate, but still it is of periodic type and it has some  a priori bounds.  So therefore, therefore, this is impossible and it implies that we cannot find such degenerations  at first place.  And that's kind of a rough kind of cartoon idea of the argument that allows to promote  non uniform bounds to uniform bounds.  So let me make like a few final comments like first, as I said, we need to work with the  digital disk, so we add parabolic fjords to the digital disk.  And we do make such construction on all scales, so to catch part of the technicality is sort  of to show that different levels don't really interact much.  So this is the way how we do it is by establishing a certain force a priori bounds.  Well, another remark that kind of maybe let me say like about applications.  So since this should be digital disk kind of exists for all data, so this would imply  that for every data there would be some HCOG, let's say for every data that is not rational  for every rational data, there will be a HCOG that contains the critical orbit.  And for that, that agree is one.  So we can't kind of claim at this stage the topology of that patch call, but at least  it would exist and the degree will be one.  So in particular, the post-critical set is not everything, which is an open question  outside of the Inno-Shishikura class.  But in the Inno-Shishikura class, so as I said, there is now full topological understanding  of such objects and she are hopefully in kind of in the future, this topological understanding  can be improved for all neutral polynomials.  Let me stop here.  Okay.  Thank you.  Are there any questions?  You are online?  Okay.  Maybe I have a question.  So Jeremy's philosophy was something like if your life is bad today, then your life  was even worse yesterday.  And your philosophy is that if your life is bad today, then it will be even worse tomorrow.  So this is a joke, but do you have a dynamical or more like serious explanation why it goes  another way?  I mean, the reason is this, Naik, Lema.  Well, in the Jeremy situation, let me go back to the Jeremy situation.  Well, the Julia sets are disjoint and he doesn't need to kind of understand kind of  the boundary of each, the geometry near the boundary of each small Julia set.  He can kind of think that each small Julia set is roughly kind of a disk for this type  of arguments.  So here, here the Ziggle disk is one object, so it is a connected object.  So we need to kind of understand the ideal boundary of the Ziggle disk with respect to  the geometry outside of the Ziggle disk.  So this kind of leads up to understanding the submergence of the white rectangles inside  the Ziggle disk.  And this sort of forces this philosophy that this kind of forces to go into the deep scale  instead of going into the shallow case, into this shallow scale.  So this is really because we need to understand the submergence of white rectangles inside  the Ziggle disk.  Yeah, let me also mention that I gave a mini course, a full lecture mini course last December  about more details of this type of argument.  So it's available on the Simon Center video collection.  More questions?  Can I ask a question?  Sure.  Yeah.  So I'm wondering if you are also able to control the geometry of the fjords within this pseudo  Ziggle disk that you talked about.  For example, the fjords start to say a spoiler on the inside.  A spoiler about the fixed point or to do other things.  Right.  Yeah, we don't.  Right.  Yeah.  So in some sense, our arguments allow us to understand the external boundary.  So our arguments sort of allow us to understand the external boundary of this pseudo Ziggle  disk.  But we can't really say much, I guess, at this stage about what's happening around the  fixed point.  So we don't.  Yeah, we can't say how fjords would spiral.  Right.  So I mean, maybe a different question is if I understand correctly, the pseudo Ziggle  disk contains, say, the first few iterates of the critical point.  Yeah.  Is this right?  Maybe could I really state your theorem that for any, like, no finite number m, there exists  a k such that for any bound, state of bounded type, there is this pseudo Ziggle disk, which  contains the first m iterates of the critical point.  Yes, this is correct.  So we may need to kind of enlarge k, the capital K, but yeah, by enlarging this absolute k,  we can control much more pseudo Ziggle disks.  We can control much more the perspective orbit.  Yes.  Right.  Because without stating it like that, because it seems to me that there could be maybe a  trivial example of a pseudo Ziggle disk, you just take a closed curve which passes through  the critical point.  Yeah.  And the map restricts to a degree warm up.  No, but.  Because your pseudo Ziggle disk is not invariant.  Yeah, but pseudo Ziggle disks does contain the original Ziggle disk.  And it is kind of an open question.  Well, I mean, it was an open question kind of why the boundary of the Ziggle disk is  not everything.  I have a question about higher degree polynomials.  So for instance, in the family of all degree three polynomials, would you also have a uniform  k such that your theorem holds?  Right.  I think, I mean, kind of what we really do.  So we start with the Ziggle disks that we understand with that are non-uniform.  And then we kind of provide a priori bounds for them.  Well now for Cubics.  I mean for Cubics, if we consider a Ziggle disk of periodic type, perhaps a non-uniform  that rotates.  And if this pseudo Ziggle disk contains both critical points on the boundary, then I would  expect that the technique should be applicable perhaps with minor modification.  So in particular, such objects kind of are compact or at least I would anticipate it.  But in the cubic case, it can may happen that one, I mean, we know that one of the critical  points must be on the boundary.  This is thanks to Joe Digi's surgery and Schmitz's show how it can be applied in the  multi-critical for multi-critical polynomials.  So if we have a Ziggle disk of periodic type, then it must contain one of the critical points.  But then another critical point must be sort of kind of on the decorations.  This may be kind of a challenge.  Because I mean, if you want to understand all neutral cubic polynomials, then we want  to understand also the parameter that are obtained by taking limits of these objects.  Okay, thank you.  Okay, let us send the speaker again.  Thank you, Jeff.  Okay, thank you.
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So we consider the space of all quadratic like maps up to a fine conjugacy. This is a nice analytic space. The renormalization operator would be iteration and restriction. Both are very analytic operations. So the renormalization operator would itself be analytic and it will be contracting. So if we have two maps that are hybrid conjugate, then the renormalizations will also be hybrid conjugate and the quasi conformal distance will be smaller. A caveat is that the dimension of this space is infinite. Well, and our goal is again to find a fixed point or more generally renormalization horseshoe. And for that, we do want some compactness property because since our space is infinite dimension, we can't go far without any compactness property. And it is often called a priori bounds in this subject. And what happens is that often a priori bounds can be established in the near degenerate regime. And this is the main topic of my subject. I think I have some echo. Okay, not anymore. I'd like first to summarize what is known about the near degenerate regime for the first time and quadratic like renormalization theory. And then I'll move to the neutral renormalization. A fundamental fact about the remand surfaces is that they degenerate along thin analyte and white rectangles. A more familiar case is the situation with the compact remand surfaces, perhaps punctured by many points. So in this case, the surface degenerates along thin analyte. So it is called the syncyc decomposition. And if we have on the other hand a remand surface with boundary, so let's assume that we have some Jordan domain in the complex plane minus finitely many Jordan domains. Then such object degenerates along finitely many white rectangles. So it is more convenient to draw not the realistic picture, but a schematic one. So let's apply Photoshop and replace all white rectangles with narrow rectangles. So the convention from now on would be that narrow on pictures means white in reality. So this is because we are interested in homotopical properties of white rectangles. In the Boston theory, when we iterate, if under the iteration we develop long analyte, then there is a certain matrix that controls the degeneration. First of all, if the degeneration develops, then it develops along invariant, multi-curve, and applying the Grotchen equality, we easily obtain an estimate from below of the moduli of the analyte of the pullback through the certain matrix and the moduli of the surface before the pullback. And in fact, in the degenerate situation, this inequality can be effectively reversed and replaced by equality perhaps with an additive correction. Well, this is because, well, our map on analyte is a covering map, and we understand what's happening with covering maps. And the map on thick parts would be belong to some compact set. And this is the idea of how we establish an invariant compact set under the certain iteration. So of course, I skipped a lot of details. In the quadratic life case, so I'm going to consider the primitive bounded case, and I will discuss the argument by Jeremy Khan. So let's consider some quadratic polynomial and just assume that we are in the degenerate situation, so the modulus between u and j is very small. So let's apply renormalization and replace the Julia set by a cycle of small Julia sets. And our task, our goal is to understand what are the white rectangles between small Julia set and the external boundary. And then try to catch the contradiction. And as we can see on the picture, there are two types of white rectangles, horizontal, that connect two small Julia sets, and vertical. And remember that in the certain case, the generation develops along an invariant multi-curve. Well, same story here, that horizontal rectangles must be developed along an invariant graph that connects small Julia sets. But it's full clear that the only invariant graph that connects small Julia set is the hybrid tree. So therefore, horizontal rectangles must be developed along the hybrid tree, but we can say even more. So if we consider the associated PCF model, or it has the hybrid tree, so we can look at the corresponding matrix. Well, and just like the certain matrix controls what is happening with long annulite in the pullback iteration. This matrix controls what's happening with the horizontal rectangles. And in particular, if we apply a grudge type inequality, and if we assume that the core entropy of F restricted to the hybrid tree is positive, then it would tell us that the horizontal rectangles are not very good. So if we assume that there are no vertical rectangles, then this would already lead to a contradiction, well, more or less the same argument as in the Kyrsten case. But a correct statement is that vertical vertical rectangles must form a substantial part of all vertical rectangles. So horizontal rectangles cannot dominate. This is a key step in the Jeremy's argument for the MLC for bounded primitive parameters. Dima, you are proving something. What are you trying to prove? What is the statement? Well, I'm trying to make an analogy between the quadratic life renormalization and Kyrsten. You're making an argument that something is a contradiction of something. What is the statement you're trying to prove? Well, the statement is that the vertical rectangles form a substantial part of all white rectangles. So if we take, so that they will exist at constant C, so that if we sum up, yes. What are these rectangles? Well, rectangles, these are rectangles that are in the same, in the composition of the Riemann surface. When we take you and she removed for the cycle of small Julia sets. So this is a Riemann surface with a boundary. So this surface has the same thing, the composition. Well, elements of the same thing, the compositions will be white rectangles. And the statement is that, well, there was a true statement is that this white rectangle that are horizontal are aligned with the cupboard tree. And the second statement is that if we sum up the width of vertical rectangles, then this would form a substantial part of all rectangles. So I'm trying to, you said we have to Photoshop the picture you're drawing. And I'm trying to do this in my brain. Let me see whether I got it right. The blue rectangles, contrary to what you've drawn, mean that this little Julia sets are much closer to each other than kind of the diameter. So this is the realistic picture. Well, this is the other kind, but the blue ones, the horizontal ones. So the idea is that the little Julia sets are closer to each other than their diameter by some order of magnitude. And so this is two boundary components. And the other boundary components are from the little Julia sets to the outer ellipse boundary, the boundary of you. And here you're saying again that the little Julia sets are closer to the boundary of you than their diameter. Yeah, you can say it in this way. And I'm trying to Photoshop my mental picture here. And then you're saying of all the area in you that we see minus the little Julia set, this is essentially filled by your rectangles. And you're saying you want to say that the definite fraction is of these rectangles is between the little Julia sets and the boundary as opposed to the blue ones between the Julia sets. Correct. Yeah. So this is the statement that Hemel wanted. I ought to be able to understand what it means. I don't know why things are being called rectangles. And I've never seen this language before. And I just don't know what it is that's being talked about. Horizontal sides on these disks or on the boundary of the big D. Assume that you have a disk with several holes, just a few holes. Your rectangle is a topological rectangle whose horizontal sides on the boundary of this Riemann surface. And then you can, so it is just a topological rectangle whose horizontal sides are on the boundary of our Riemann surface in topological and on trivial guys. And the statement is the degeneration of such Riemann surfaces happens when this rectangle, some of these rectangles become white. And so that is the issue of renormalization theory as Dima was trying to convey is to prevent this degeneration, this kind of degeneration. And it is in a way very much similar to how Sursan prevents degeneration along long tubes in the case of compact Riemann surface of finite type. So, okay. Well, in fact, this is what I wanted to say about the quadratic like renormalization. So namely, I wanted to make this parallel that if you look at what's happening, then the argument is indeed kind of very close to what's happening in the Sursan iteration. And that's kind of one of the key steps, if not the key step in Jeremy Khan proof of the MLC in the primitive bounded case. An important tool is the covering lemma developed by Jeremy and Misha. And in the follow up work, Misha and Jeremy push the theory much farther. So now it handles quite well the primitive case, assuming that we are kind of uniformly away from the molecule. So roughly speaking, assuming that the core entropy is at least some epsilon. So roughly what they did, they translated all the theory of the primitive renormalization into the language of the near degenerate regime. Yeah, and establish a priori bounds under such kind of condition. Okay, so more details will be tomorrow in Misha's talk. So this is just before the bulliness. So please come to bulliness. So let me also make a comment about the Sursan theory. So there is another approach to the renormalization to the realization problem by Dylan. So he was talking about it yesterday. And in some sense, the near degenerate regime is even more explicit in his situation. So he starts with a map between graphs. So then he replaces graphs with a small neighborhood. This will be a very degenerate remand surface, and then he does the realization for this degenerate remand surfaces. And then after that, he does the surgery and obtains epispirically finite rational map. So Kevin H. Dylan will be given a Zoom mini course in November on the renormalization and dynamics seminar. So everyone who is interested to the subject is welcome to join. Okay, good. So much now. I'd like to go to the near neutral renormalization theory. So this is the theory associated with the boundary of the main capoboli component. So here are some examples of maps. So all of them are digital disks, but some of these digital disks are degenerate. So this is the golden digital disk. So this digital disk is a quasi disk that rotates around the center. So this digital disk has the rotation number that is very close to zero. So it is still a quasi disk, but the center of rotation will be very close to the boundary. So here, the rotation number is close to one half. And then the digital disk will be like this. So it will start to develop parabolic fjords. But still it's kind of a nice object. So as I will talk later. So here the rotation number is close to one quarter. And we can see that the digital disk has four parabolic fjords. So it rotates. So it's not, well, it wants to degenerate and it develops parabolic fjords. So there are several reasons, or there are many reasons why we might be interested in the neutral renormalization. First, the neutral renormalization controls the quadratic like renormalization near the cardioid. So this is where the primitive renormalization theory fails. It requires certain adjustments. In particular, there is no good theory of puzzles near the boundary of the cardioid. Well, also, near neutral renormalization is very much related to the theory of near-circle maps and such maps were in the focus for a long time. Well, perhaps even from the beginning of the dynamical systems, when the dynamical systems were motivated by the celestial mechanics like planets moving around the sun. Well, and since we like the transcendental dynamics, at least on this conference, well, a good news is that the transcendental dynamics also emerges on the unstable manifolds. So this is kind of one way how we can understand combinatorics of neutral renormalization. Namely the combinatorics can be described using this particular map. So this is the Julia set of this cubic polynomial. So it has a parabolic point here, and it attracts one critical point, and another critical point is here, and it maps to the fixed point under one iteration, S221. So this field, Julia set, is very similar to the molecule, and in fact, if we endow the molecule with the Brunner DOD surgery, and I will refer to the resulting map as the molecule map, then this should be conjugate to this topologically. Well, in fact, this is equivalent to the MLC conjecture, the satellite case, that these two dynamical systems are topologically conjugate. And the Brunner DOD surgery, well, let me kind of describe the idea on the case with rabbits. So if we have a peak of a Q-rabbit, so it's the rabbit that rotates with this rotational number, then assuming that P is smaller than 2 over 2, we can remove dynamically PES from the rabbit. So we do the surgery. Well, and then after the DOD Brunner surgery, we obtain another rabbit, which will have the rotation number P over Q minus P. Well, if P is bigger than Q over 2, then while we symmetrized the construction, well, we also map the basillic apabolic component into the main apabolic component using the basillic straightening map. So everything assembles into a continuous map around the main molecule, and the MLC conjecture of the satellite case say that this must be topologically conjugate to this molecule. And periodic points on the main apabolic component of the molecule map are exactly the legal maps of periodic type. Alternatively, we can describe the legal maps of periodic type using continuous fraction expansion. So this is the periodic point of the smallest period, so here is period 2 of the molecule map. So this periodic cycle is formed by the golden-ziggle parameters, so the upper golden-ziggle parameters and the lower ziggle parameter. Once we have a periodic point, we should expect self-similarity because around periodic points we may want to linearize everything. And this is indeed what happens with the Mandelbrot set. So if we zoom in at the golden-ziggle parameter, then we see self-similarity. So this zoom is almost indistinguishable from the zoom next level. So this is called universality. The combinatorial reason is because the golden-ziggle parameter is a periodic cycle. But of course, in order to justify this self-similarity properly, we need to develop the randomization theory. Let me also make a remark that if we consider some other parameter space, for example, the space of quadratic rational maps, where one of the critical points is the periodic, and if we zoom in at the corresponding golden-ziggle parameter, then again this picture becomes almost indistinguishable from the corresponding picture of the Mandelbrot set. And then the scaling will be with the same universal constant lambda star. So here would also be. And in fact, both of these pictures converge to the limiting transcendental dynamics. And the limiting transcendental dynamics can be used in order to identify these pictures well at least partially, but hopefully in the future there would be a complete identification. Okay, so this is a summary of the neutral randomization theory. So the first conjectures appeared in the 80s. So that was the time when many physicists discovered many interesting facts. So mathematicians call them conjectures. Kurt McMullen constructed randomization ziggle periodic points associated with ziggle disks of a periodic type. He also showed the exponential contraction. So Ina and Shishikura proved the publicity of the neoparabolic randomization. So this randomization controls what happens in the neighborhood of a parabolic point. And this was important for the theory because the publicity really allows to study nearby maps. And what they did, they proved the publicity for the actual parabolic map. And since the publicity is an open condition, it also holds in a neighborhood and it allows to study maps that were out of reach before. For example, there are many applications to the area problem. There is now kind of almost a full or complete topological understanding of the hedgehog of a Kramer parameter in the Ina Shishikura class. So Kramer polynomials turns out to be not that complicated after all. Well, there are applications to the MLC conjecture in the satellite case and also ergodic properties of neutral polynomials. So the original argument by Ina and Shishikura involved some computer assistants are not later presented a computer free proof and she also works for unicritical maps also high type. Denise Guidershuk and Michael Jampolsky presented a computer assistant proof for the actual golden mean, the digital disk, namely for this main periodic cycle. Well, then in a joint work with Mishan Nikita, we proved the publicity for bounded type parameters while we involved the packman renormalization. Well, again, the publicity means that we can study nearby maps. So we understand quite well what is happening with periodic, with digital disk or periodic type. And once we have publicity, we can study also nearby maps, and we obtain some applications to also scaling and area problems. And we constructed the first example of the MLC of the satellite bounded type. Let me also mention briefly the Geodigis surgery. So this is a way how we understand digital disks of bounded type. Otherwise, we can't control the critical orbit. So the idea is that we start with the Blaschke product, a Blaschke product restricts to a circle map. So it's going to be a critical circle map. And since the critical circle maps are quite well understood, in particular, if our rotation number is of bounded type, then the restrictions to the circle will be QC conjugate to the rotation. Then this allows to apply a quasi conformal surgery and obtain out of the Blaschke product a Ziegler polynomial of periodic type. So we forget the dynamics inside the unit disk and we do the surgery, and this is an essential bit for the surgery. And what's important is that we obtain a map where we control the past critical orbit. The past critical orbit would be rotated on the boundary of the Ziegler quasi disk. But there are limitations for this surgery because the surgery cannot produce a framework polynomials. Okay, so let me now state Ethereum, so it's a joint work with Misha. So the theorem states that there are uniform a priori bounds for all neutral polynomials on the boundary of the capoboli component. So there are different ways of stating it. So let me first state kind of a rough version. So a rough version claims that when the Ziegler disks of a bounded type degenerate, then they can degenerate only in a star-like way. So as we can see on the picture, so the degenerations has this type, but we don't see the degenerations like this. So this is we don't see on pictures. And this is the statement, so it can be made precise, but roughly speaking, when a Ziegler disk degenerate, then this is allowed. So such degenerations is allowed, and it can also happen on all scales. So this is like scale, let's say one, and if we zoom in, then we would see more parabolic fjords, and still there would be star-like patient. And this is not allowed. So a bit more precise way of stating it is that for any rotation number of a bounded type, but M can be any Ziegler disk, so this is our Ziegler disk that rotates. This Ziegler disk can be put inside a quasi-disk with a uniform dilatation, such that the degree of the map restricted to this enlarged quasi-disk, so we call it pseudo-ziegler disk. So such that the degree is one of the map restricted to this pseudo-ziegler disk. And as a corollary, because quasi-conformal objects are compact, such pseudo-ziegler disk exists for all neutral maps, even for Kramer polynomials, so there would be a pseudo-ziegler disk that is sort of almost invariant, and this pseudo-ziegler disk would have an absolute dilatation. So these pseudo-ziegler disks are obtained by adding these parabolic fjords on all scales. So this is how we construct the Ziegler disk. Dima, it's not really visible. So Raph is speaking out of a star. We can produce a... Dima, could you... Accuracy. Dima, could you please repeat your drawing, because it was not very visible. It's all dark. Is it better now? Are you talking about this? Yes. Maybe you can make it closer to us. Right. So is it better? Yes. Is it? Right. So what I'm saying is that pseudo-ziegler disk can be obtained by adding such parabolic fjords, while in stimuli with other periodic Ziegler parameters. So whenever a parabolic fjords is developed, we can add it to the object. So I was saying it more schematically. If we have a star, then we can add fjords between star and obtain a uniform QC disk. Right. And another way of stating a priori bounce is that there would exist a fiber-wise compact renomization operator, cylinder renomization operator around MacMallon periodic points when we take the closure of MacMallon periodic points. So this is renomization horseshoe. So we take all MacMallon periodic renomization points. We take the closure and the statement is that this will be a compact set of maps and there would be a renomization operator around this compact set of maps. So any questions about the statement? So any questions at this stage? Before I'll try to give some ideas of the proofs. Okay, good. So as I said, the proof is in the near-degenerate regime and here is like one basic idea that suppose we have a single disk that rotates. And suppose we have two intervals, i and j. And suppose that there is a vitric tunnel that connects them. So again, this is kind of a photoshop, but in reality this vitric tunnel will be like this. Well, as now, if combinatorially we can pull back these two intervals, i and j, two intervals located like this, so between them. And if we assume that the critical value is not on the way, then we can pull back this vitric tunnel and obtain a vitric tunnel that cross intersects the original vitric tunnel. And this is impossible because vitric tunnels do not cross intersect. And this type of arguments allows to cover like a lot of cases. So a good half of all the arguments are based on this principle. So in particular, this type of arguments allows to understand the geometry of parabolic fjords. Let me now kind of try to give a cartoon proof of the logic. So this will be like a cartoon rough presentation of the main logic of the argument. Assume that we don't have uniform a priori bounds. So this means that we have a digital disk that looks like this in reality. So like one peninsula approach becomes very close to another peninsula. Then there is a vitric tunnel between these two peninsulas. So as I said, we prefer to do schematic pictures. So if we apply the Photoshop, then we obtain like we put our stuff back, but then we would have a vitric tunnel between these two peninsula. So vitric tunnels geometrically looks like this. Well, it's now this degeneration can be spread around. So the digital disk rotates. The covering lemma allows to kind of spread this degeneration around to more or less all the peninsula, not more or less, but to all peninsulas. But now if we try to spread this rectangle to a rectangle that connects these two peninsula, then this is impossible because vitric tunnels cannot cross intersect. So therefore, when we apply the covering lemma and move the original rectangle to the rectangle that connects this two peninsula, the new rectangle must go through the digital disk. So it must be submerged. Well, and this submergence could be understood because inside the unit disk we have the complex structure of the standard disk that rotates. So in particular, these degeneration can be localized. So more of the degeneration would go like this. And like only a big of one of the degeneration would be beyond, well, would be below this. We call these techniques like snakes. And we have several versions of the snake lemma. Right. But now after localizing the degeneration, we can also observe that roughly this degeneration fits into two rectangles. And one of these rectangles should be twice wider than the original rectangle. So this allows kind of us to say that either this rectangle or this rectangle will be twice bigger than the original rectangle. So we can obtain from a degeneration, bigger degeneration. Well, and then this bigger degeneration could be put inside a fjord of a, inside the epininsular of a deeper scale. But here comes sort of the main logic that if we originally had a big degeneration, then we can find bigger and bigger degeneration at its depth and we can construct like a sequence of rectangles so that each rectangle I n will be to the power n wider than the original rectangle. But such sequence cannot exist because, because we do have a non uniform a priori bounds. So our digital disk was of periodic type. So it might be kind of super degenerate, but still it is of periodic type and it has some a priori bounds. So therefore, therefore, this is impossible and it implies that we cannot find such degenerations at first place. And that's kind of a rough kind of cartoon idea of the argument that allows to promote non uniform bounds to uniform bounds. So let me make like a few final comments like first, as I said, we need to work with the digital disk, so we add parabolic fjords to the digital disk. And we do make such construction on all scales, so to catch part of the technicality is sort of to show that different levels don't really interact much. So this is the way how we do it is by establishing a certain force a priori bounds. Well, another remark that kind of maybe let me say like about applications. So since this should be digital disk kind of exists for all data, so this would imply that for every data there would be some HCOG, let's say for every data that is not rational for every rational data, there will be a HCOG that contains the critical orbit. And for that, that agree is one. So we can't kind of claim at this stage the topology of that patch call, but at least it would exist and the degree will be one. So in particular, the post-critical set is not everything, which is an open question outside of the Inno-Shishikura class. But in the Inno-Shishikura class, so as I said, there is now full topological understanding of such objects and she are hopefully in kind of in the future, this topological understanding can be improved for all neutral polynomials. Let me stop here. Okay. Thank you. Are there any questions? You are online? Okay. Maybe I have a question. So Jeremy's philosophy was something like if your life is bad today, then your life was even worse yesterday. And your philosophy is that if your life is bad today, then it will be even worse tomorrow. So this is a joke, but do you have a dynamical or more like serious explanation why it goes another way? I mean, the reason is this, Naik, Lema. Well, in the Jeremy situation, let me go back to the Jeremy situation. Well, the Julia sets are disjoint and he doesn't need to kind of understand kind of the boundary of each, the geometry near the boundary of each small Julia set. He can kind of think that each small Julia set is roughly kind of a disk for this type of arguments. So here, here the Ziggle disk is one object, so it is a connected object. So we need to kind of understand the ideal boundary of the Ziggle disk with respect to the geometry outside of the Ziggle disk. So this kind of leads up to understanding the submergence of the white rectangles inside the Ziggle disk. And this sort of forces this philosophy that this kind of forces to go into the deep scale instead of going into the shallow case, into this shallow scale. So this is really because we need to understand the submergence of white rectangles inside the Ziggle disk. Yeah, let me also mention that I gave a mini course, a full lecture mini course last December about more details of this type of argument. So it's available on the Simon Center video collection. More questions? Can I ask a question? Sure. Yeah. So I'm wondering if you are also able to control the geometry of the fjords within this pseudo Ziggle disk that you talked about. For example, the fjords start to say a spoiler on the inside. A spoiler about the fixed point or to do other things. Right. Yeah, we don't. Right. Yeah. So in some sense, our arguments allow us to understand the external boundary. So our arguments sort of allow us to understand the external boundary of this pseudo Ziggle disk. But we can't really say much, I guess, at this stage about what's happening around the fixed point. So we don't. Yeah, we can't say how fjords would spiral. Right. So I mean, maybe a different question is if I understand correctly, the pseudo Ziggle disk contains, say, the first few iterates of the critical point. Yeah. Is this right? Maybe could I really state your theorem that for any, like, no finite number m, there exists a k such that for any bound, state of bounded type, there is this pseudo Ziggle disk, which contains the first m iterates of the critical point. Yes, this is correct. So we may need to kind of enlarge k, the capital K, but yeah, by enlarging this absolute k, we can control much more pseudo Ziggle disks. We can control much more the perspective orbit. Yes. Right. Because without stating it like that, because it seems to me that there could be maybe a trivial example of a pseudo Ziggle disk, you just take a closed curve which passes through the critical point. Yeah. And the map restricts to a degree warm up. No, but. Because your pseudo Ziggle disk is not invariant. Yeah, but pseudo Ziggle disks does contain the original Ziggle disk. And it is kind of an open question. Well, I mean, it was an open question kind of why the boundary of the Ziggle disk is not everything. I have a question about higher degree polynomials. So for instance, in the family of all degree three polynomials, would you also have a uniform k such that your theorem holds? Right. I think, I mean, kind of what we really do. So we start with the Ziggle disks that we understand with that are non-uniform. And then we kind of provide a priori bounds for them. Well now for Cubics. I mean for Cubics, if we consider a Ziggle disk of periodic type, perhaps a non-uniform that rotates. And if this pseudo Ziggle disk contains both critical points on the boundary, then I would expect that the technique should be applicable perhaps with minor modification. So in particular, such objects kind of are compact or at least I would anticipate it. But in the cubic case, it can may happen that one, I mean, we know that one of the critical points must be on the boundary. This is thanks to Joe Digi's surgery and Schmitz's show how it can be applied in the multi-critical for multi-critical polynomials. So if we have a Ziggle disk of periodic type, then it must contain one of the critical points. But then another critical point must be sort of kind of on the decorations. This may be kind of a challenge. Because I mean, if you want to understand all neutral cubic polynomials, then we want to understand also the parameter that are obtained by taking limits of these objects. Okay, thank you. Okay, let us send the speaker again. Thank you, Jeff. Okay, thank you.
I'll move to the neutral renormalization. A fundamental fact about the remand surfaces is that they degenerate along thin analyte and white statement is that vertical vertical rectangles to the realization problem by Dylan. So he was in his situation. So he starts with a map between November on the renormalization and dynamics associated with the boundary of the main capoboli be like this. So it will start to develop the dynamical systems were motivated by the unstable manifolds. So this is kind of one way how we can understand combinatorics of neutral of the Mandelbrot set. And then the scaling will then in a joint work with Mishan Nikita, we with Misha. So the theorem states that there are uniform a priori bounds for all neutral proof is in the near-degenerate regime and here this is the way how we do it is by establishing a