 Thank you very much. Like to thank the organizers is a pleasure, the great, is always a great pleasure to come to Trieste. Think like many people said, it also was one of my first international conference. So it's always a great pleasure being here. So I'll talk about some works in progress, about trying to give a global picture, of a global description of the fractal geometry, of the fractal geometric properties of a typical Hochschul, a typical hyperbolic sets, at least of a typical Hochschul in arbitrary dimension. So for a dissipative dynamical system, so you know that many dynamical systems are not uniformly hyperbolic, but it's quite often even dynamical systems which are not hyperbolic contain subsystems which are hyperbolic. Typically, they contain a lot of periodic orbit and in many cases, non-trivial invariant sets contain non-trivial hyperbolic sets. In particular, quite often non-trivial invariant sets by dynamical systems contain Hochschuls, contain hyperbolic sets with zero topological dimension, but which can have, say, large fractal dimension. And we try to understand the geometry of such hyperbolic sets of Hochschuls as well as possible for typical, say, infinity dynamics. So it's a talk about uniformly hyperbolic dynamics, but about some geometrical problems which are still not very well understood, for instance. Some very basic questions, for instance, about the typical continuity of fractal dimensions of Hochschuls or of stable and unstable control sets associated with Hochschuls are still open and what are the good notion of fractal dimensions we should use for typical Hochschuls? It's also not completely clear in general, so we'll try to give some answers to these questions. So let's start with a very well-known case of dimension 2, of Hochschuls dimension 2, which are important to understand bifurcations of hyperbolic dynamics in dimension 2. So at least personally, I began studying this subject motivated by the study of homoclinic bifurcation in dimension 2, so I'll recall very briefly some of the history of this problem. So here we have a Hochschul there. Let's see how it works. It has a light somewhere. See? No. Pardon. Oh, no. Sorry. Here. Yeah. This one. OK. Pardon. Voila. OK. So this is a Hochschul, and here we create a first homoclinic tangency associated to the Hochschul, and then we unfold the tangency and to see what happens for small positive values of the parameter. And this is what happens after bifurcation. We may create a lot of new tangencies. Well, it depends a lot on the geometry of the original Hochschul. Well, indeed, new tangencies essentially correspond to intersections of contour sets in a real line. So if you extend these stable and unstable laminations to filiations defined at the neighborhoods of them, then we have a line of tangencies. So a line where the laminations can have some non-trivial tangences, this line here. And so if we consider these holonomies, we have two contour sets, a red contour set and a blue contour set, so a stable, so the image of stable and unstable contour sets in this line, which are moving when we unfold the bifurcation. And we want to understand whether these contour sets intersect often or not. And this is, in first approximation, is related to understanding the size of the arithmetic difference of these two contour sets. If one of the two contour sets is fixed and the other moves with constant speed, the set of parameters for which they have nonempty intersection is the arithmetic difference. And if the rock shoe is small, so if the house of dimension of the rock shoe, which in dimension 2 is the sum of the house of dimension of both contour sets, if it is smaller than 1, then this arithmetic difference is a small set, is a set of house of dimension smaller than 1 and of zero Lebesgue measure in this line. So it means that for most parameters, we don't have intersection between this contour set. And indeed, it implies this fact was used by Jacques and Floris, by Paulis and Takens, to show that in a typical homo-clinic bifurcation associated with small rock shoe to rock shoe with dimensions smaller than 1, indeed with full Lebesgue density at the initial bifurcation parameter, there is hyperboleicity. So there is prevalence of hyperboleicity at the initial bifurcation parameter. So new homo-clinic tangences are very rare in this situation. On the other hand, when the house of dimension of the rock shoe is larger than 1, then the arithmetic difference typically has nonempty interior. So typically, you have nonempty intervals of persistent tangences. And indeed, this corresponds to open sets in the parameter with positive Lebesgue density corresponding to new house phenomenon, to persistent homo-clinic tangences. So the geometry of the rock shoe is very much related to the bifurcation diagram to understanding the prevalent dynamical behavior after the first homo-clinic bifurcation. So a main problem here was to understand these arithmetic differences, or more precisely, these intersections of contour sets in the real line. So indeed, this result about the case when the sum of the house dimension is larger than 1 was a joint work with Yokoz in 2001. So we showed that typically when the sum of the house of dimensions of two regular contour sets in the real line is larger than 1, then almost every position of intersection is a position of stable intersection, which means that if you move a little bit the pair of contour sets, or even if we change a little bit the dynamics which define the contour sets, they still have nonempty intersections. So they have like a transversal intersection in some sense. It's some sort of fractal transversality phenomenon. And in other words, we show so we can reformulate this result with Yokoz in order to show that if you have a Hokushu with dimension larger than 1, then typical projections of the Hokushu to the real line, or typical images of the Hokushu by differentiable functions, have typically nonempty interior in the real line in a persistent way. So if you perturb a little bit the function in the C1 topology it still has nonempty interior. And this is open in the C1 plus epsilon topology in the Hokushu. It's dense in the C infinity topology and open in C1 plus something topology, C1 plus epsilon for some positive epsilon. But it's not open in the, so this result is not true in the C1 topology. For typical C1 Hokushu, the projections will be counter set. But this is another story. But this is a typical phenomenon in higher topology. So it's dense in C infinity topology. It's open in C1 plus epsilon topology. So it's a reasonably good result in this context. But it's not, so it's C1 open in the projection. So the map which projects in the real line can be C1 and can be perturbed in the C1 topology. But not the Hokushu. We need the dynamic to be at least C2 in this context. But in the results I will describe later, C1 will be enough. But so what happens about the geometry of Hokushu in dimension 2? Well, the house of dimension of Hokushu in dimension 2 is equal to the sum of the house of dimensions of the stable and unstable counter sets. A stable counter set is an intersection of the Hokushu with a local stable manifold. And an unstable counter set is an intersection with a local unstable manifold. And well, it depends on the choice of the manifold. But indeed, all these intersections are defiomorphic because the holonomies are nice. The holonomies are at least C1. Indeed, C1 plus something, it's 1 plus epsilon if the Hokushu is at least C2, for instance. So the situation is quite nice in the geometric point of view. The house of dimension depends continuously, even in the C1 topology of Hokushu's in dimension 2. OK, but the situation, so this talk is about higher dimension, dimension higher than 1. But in some sense, this result with your cause gives some tools which at least psychologically are important in the development of the results I will describe in higher dimension. So at least the structure of the proofs, so the technique of the proofs are clearly inspired in the proof of this result with your cause. And indeed, we have two main ingredients in the proof of this theorem which solved the conjecture by Pajaco-Pales, that typically for counter-sets on the real line. So if the dimension is smaller than 1, then the arithmetic difference is very small. It has 0, a big measure. And if the sum of dimension is larger than 1, then they have a lot of stable intersections. The arithmetic difference typically has no empty interior. So we have a quite precise dichotomy in the sense for most pairs for an open-ended set of pairs of regular counter-sets on the real line. So the main techniques in the proof of these results are first the establishment of a criterion, also of a sufficient condition, which implies stable intersection, which is sufficient to guarantee that two counter-sets on the real line have an intersection that they cannot destroy by changing a little bit the pair of counter-sets. And then we need to show that this condition is satisfied by most pairs of counter-sets whose sum of dimension is larger than 1. And here we use a version of Erdos' probabilistic method. In the sense that, well, we have two C infinity counter-sets on the real line, we'll make a perturbation to one of them, fix one of the counter-sets, and make a perturbation to the other one in order to create stable intersections. Indeed, in order to the new pair of counter-sets to satisfy these recurrent compact sets criterion that we introduced before. So these perturbations are not quite explicit. Indeed, what we do is to create a family of perturbations with a very large number of parameters and to prove that for most parameters, the perturbation works. So for most parameters, indeed, we are creating a recurrent compact set, and everything will work. But so this, in some sense, this ingredient, so this philosophy, we will still be present in the results I will describe now. So let's start discussing the situation in a higher dimension, dimension larger than 2. And we will focus on the study of the Hochschule themselves, not on bifurcation. So perhaps later, we can use some information to get here to study some more. But here, we are concentrating in understanding geometry of Hochschule. So we already have a problem in dimension larger than 2. Since, for instance, the house of dimension of Hochschule is not always continuous, even in the infinite topology, in dimension larger than 2, indeed, what happens is that a stable contour set, so you can have a contour set given by contractions, which can be a stable contour set of a Hochschule, which can be something like this. We have a, let me try to make a good picture. I have two things like this. So yes. So we may have, say, a square, which is sent to the interior of itself by two contractions, two affine contractions, say, F1 and F2. By two rectangles, we have two invariant directions, the horizontal and vertical directions. And the horizontal projections of these two rectangles is essentially exactly the same interval. So if we iterate this, the horizontal projection and the limit will be just one point. But if we move a little bit this picture, so this will be a skew product. You'll have a picture like this. So in the interval here is like, and we have two maps which coincide like this. But here, we perturb, we have two slightly different maps, which slope smaller than one, but close to one, much larger than one half. And here, we can have an invariant interval by a set of maps like this. So here, by small perturbations, the projection can contain an interval. But here, the projection can be a point. And in the vertical direction, if the contraction is large here, the dimension can be arbitrarily small, so it can be much smaller than one. So here, we may have a jump of the dimension of the invariant set here from something which is very small, smaller than one half, say, to something which is at least one, because the projection will contain an interval. So this is a rough idea of an example. There is a family of examples like this. But on the other hand, this example is very special because I have two affine maps. So you have a skew product such that the projections are the same interval exactly, so the same map. So this seems to be not a generic phenomenon. But anyhow, the house of dimension of Hochschule in dimension larger than three is not always continuous. But it's a reasonable question to ask whether it's a generically continuous. Well, Alina joined work with Giacomo and Marcelo in order to understand better homoclinic bifurcations in dimension larger than two. We introduced a fractal invariant, so a fractal dimension which we called the stable dimension. And analogously, we have an upper unstable dimension associated to the unstable counter set, which now we generalize it. So I will present a family of fractal dimensions, of new fractal dimensions. All of them will be upper bounds for the true house of dimension of the stable counter sets. So we have the same thing for the unstable counter sets. So yeah, so the idea is to work with vertical cylinders. So we have a Hochschule, and in the Hochschule we have an initial Markov partition. And we have some vertical cylinders. And I'm not good at drawing things, but let's try to. So how is the typical picture in the square? So in the square, we have, say, some horizontal strips. In the easiest case, we have two horizontal strips, which are sent to vertical strips like this. So the vertical cylinders are corresponding to these vertical strips are image. And we have horizontal strips like this. When we iterate the dynamics here or here, we will have a large number of very thin, but an exponentially large number of vertical cylinders. So we have a lot of small vertical cylinders. Well, the geometry of vertical cylinders, well, they are thin. So if you take a cross section here, so the volume of the cross section is always exponentially small. Indeed, the volume does not change much. So there is a bounded distortion of the volumes. But the shape of the cross section, in principle, can change a lot. So you may have some cross sections looking like a circle, and other cross sections look like very distorted ellipses, for instance. And they need to control this. So maybe need some lemmas that show that for most cylinders, we have reasonably good control of the distortion between cross sections. But what we do in order to estimate the dimension of the, so in general, the Holonom, if you take two cross sections in the final thing, we take two stable manifolds. We have here two cantorsets, and we have a Holonomy. But in general, the Holonomy is only real continuous, not in general even ellipses. So in principle, the house of dimension can be very different of the two cantorsets. The idea is that we want to understand regular cantorsets, which are intersections of the Hochschule with local stable manifolds. And if you consider a set of vertical cylinders of some step of the construction of the Hochschule, they give natural dynamical coverings of regular cantorsets. So here we have a dynamical covering. And here we have a corresponding dynamical covering by the same cylinders. But the section of a cylinder here and here can have different shapes. So here we can have a circle here at the start of ellipses. So the Holonomy a priori is not very good. So it's only held. OK. But we have these vertical cylinders. And the idea is to each cylinder, we take this for. So let's recall what was the upper stable dimension we introduced earlier with Jaco and Marcelo. In this case, we consider just the diameter of this vertical cylinder. For each cylinder we intersected with local stable manifolds. We computed the diameter of this intersection and took the supremum of this diameter over the whole vertical cylinder. So take the maximum diameter over the cylinder. And well, this gives an upper bound for a covering. Because in each local stable manifold, we have a covering by, well, we may replace these pieces by a ball whose diameter or whose radius is this maximum diameter here. So we have a covering which gives an upper estimate for the dimension at this level. The idea is that if we take this diameter, so this is indeed the maximum diameter. PS1, here we take the first singular value of the derivative, which gives essentially the diameter. So we take the maximum diameter. So this is a dimension formula. You compute the exponent such that this sum is equal to 1. This gives an upper estimate for the house of dimension and indeed for the box dimension of all the stable counter sets at the same time. Well, in general, if the dimension of a regular counter set is larger than 1, we need to do better. Well, this is a good estimate for the house of dimension if the dimension of the stable counter sets is smaller than 1. Indeed, it's possible to prove that typically if the stable dimension is smaller than 1, then this upper stable dimension will typically coincide with the house of dimension of the stable counter sets, which will be indeed the same for all these. Indeed for typical actual value, we need some extra results that will come later. But indeed, at the end of the story, this first or this original upper stable dimension, not only are good upper estimates, they are technically simpler to compute than the house of dimension, where we need to consider arbitrary coverings by small balls. And indeed, typically they coincide with house of dimension. They are always an upper bound for house of dimension and they coincide with house of dimension, provided the house of dimension is smaller than 1. But if the house of dimension is larger than 1, then we need to refine this dimension. And we do this here. We need to work. What happens is that we are dealing with non-conformal dynamical systems. So typically, a slice of a vertical cylinder do not look like a, let's begin with, say, dimension 3 when the stable manifold has dimension 2. So a slice of a vertical cylinder looks like an ellipse, which is typically not a circle. It's typically like a very distorted ellipse like that. So we have a lot of ellipses like that. And what we do when we consider discoverings by balls with the same diameter is that we are replacing the ellipses by balls with the same diameter. Apparently, we are losing information, but this is good, provided the dimension is small. So if the whole set is small, it projects typically to a set of dimensions smaller than 1, then this is a good strategy. But if the dimension is larger than 1, then it's better to cut the ellipses in smaller pieces along the, to begin with, along the direction of the second largest, or along the second smallest contraction. So the diameter is in the direction of the smallest contraction. And here we will look to the direction of the second smallest contraction. And the second idea is to cover these by, so you divide in one extra direction and consider coverings like this. So the second, the dn2, for instance, will take the two first singular values of the derivative. So lambda 1 and lambda 2. And the idea is that this ps2, for instance, is the product of the first two singular values. So we have, in general, a formula. So what happens if the, we do an inductive definition. If the sj of lambda is smaller than j, then we stop. So it's here. If the sj of lambda is smaller than j, say, in the case of j equal to 1, we discuss this. If the dimension is smaller than 1, then all the dimensions will be the same. We'll be the s1, which is the upper stable dimension. We are happy. All the dimensions for all j will be the s1. If the dimension, so if the s1 is larger than 1, then we'll consider the s2. And what will be the s2 in this case? Well, we'll do this. We'll take each piece and divide. So each piece is essentially like a rectangle, lambda 1 times lambda 2. And we take covering by, say, lambda 1 over lambda 2 squares of size lambda 2. And what is the estimate for the dimension? It's something like sum over lambda 1 over lambda 2 times lambda 2 to the dimension s should be equal to 1. So this can be written sum of lambda 1 times lambda 2 s minus 1 equal to 1, or sum of lambda 1 2 minus s lambda 1 lambda 2 s minus 1 equal to 1. Indeed, in general, we can write these things. And this lambda 1 lambda 2, this is p1. So this is sum of pi 1 to minus s pi 2 s minus 1 equal to 1. This corresponds to this formula with j equal to 1. So pi 1 to minus s, well, s here is this d. So d is defined to be the value of s such that this sum is equal to 1. And well, then we take iterates of this. We take coverings by cylinders of higher steps of the construction. And in the limit, we show that the limit always exists. And this defines this family of upper stable dimensions. So all these dimensions are refined estimates for the true house of dimension of the hot shoe. All of them are larger than the house of dimension. And what we expect is that at the end, the last one should be typically equal to the house of dimension. And indeed, we prove this. So the main theorem is that at the end, that typically the SK will coincide with HD. But at this point, the main theorem is that for a typical hot shoe, we may find a sub-hot shoe with almost the same upper dimensions which we've defined here, which has strong stable foliations of all the dimensions we want. So well, first we have this remark. All these dimensions are upper bounds for the house of dimension and indeed for box dimension. So if some of this dimension is smaller than for some error, that the error dimension is smaller than r, then if you take a C1 image of any stable contour set to the Euclidean, to a manifold of dimension r or to the Euclidean space of dimension r, the image will have zero Lebesgue measure. It will have in this house of dimension at most this number, which is smaller than r. So if you want that some projection to an Euclidean space of dimension r or to a manifold of dimension r to have no empty interior, for instance, then necessarily the dimension should be at least r. So this is a necessary condition. And we prove some results under the hypothesis that this dimension is larger than r. OK. So what is the statement of this result? This result gives us a general picture is a mess because we may have points of periodic points with complex eigenvalues, where locally if you take, so if you look locally to the cylinders that are converging to this point, they converge like circles or like spheres. And in other points, you may have real eigenvalues where you are tending to the points like exponentially deform and ellipses, et cetera. So the geometry of a Horsch-Schul is not homogeneous at all if you take a general Horsch-Schul. What this result tries to do is to replace a general Horsch-Schul by a more homogeneous object without losing much mass in some sense, without losing much information on the geometry. So the statement is that given a typical dissipative Horsch-Schul, we may find a sub-Horsch-Schul with almost the same fractal dimensions, and which has strong stable filiations of all co-dimensions. So for each there are no periodic points with complex eigenvalues, so all the eigenvalues are real and are quite different. They have a largest eigenvalue than a second largest eigenvalue than a third largest eigenvalue. And they have dominated the compositions of all intermediate dimensions. So this theorem says that we may find a sub-Horsch-Schul at least for perturbation. Of course, we can begin with a very conformal object, but for a typical z-infinity perturbation, we may find a sub-Horsch-Schul with almost the same fractal dimensions, and with good strong stable filiations. And strong stable filiations live inside the stable leaves. And the good news is that the strong stable filiations which live inside stable leaves are differentiable, are at least C1 plus epsilon for some positive epsilon. And they will be important in the second. So the idea is to use this structure. Now we'll begin with a Horsch-Schul. And now what I can do with the remaining ties to give a very impressionistic view of what happens is to begin with a Horsch-Schul with these strong stable filiations and put a reasonable hypothesis on the dimension. So we assume that, say, the stable counter-sets have the natural dimension larger than r. So the air fractal dimension we introduced before. The upper stable dimension is larger than r. In particular, these holes, when the house of dimension is larger than r. If you assume this, then typically we have what is called an air-co-dimensional blender. So we'll show that perhaps after a small infinite perturbation, the images of stable counter-sets by typical C1 maps to the earth-dimensional Euclidean space, sorry, here is air, should be r. So the typical C1 maps to r-dimensional manifolds persistently should have non-empty interior. So if you have enough dimension, if your stable counter-sets have house of dimension larger than r, then typical images by C1 maps to manifolds of dimension r should have non-empty interior. So this correction, here is r. So this is a generalization of the previous work with Wildstone-Silval. It comes from Wildstone-Silval. This is also known as Zulu. And well, we begin with these exfoliations. And the idea is to introduce a compact recurrent set criterion. So we have the notion of renormalization operator. In this context, the idea is that we have a wall here. This H is a wall. Here we have a lot of stable leaves. I have the stable leaves of the Hock-Schul. The wall is something whose intersection with each stable leaf has dimension r. So it's transversal to the co-dimension r, strong stable foliation. So locally, it looks like a manifold of dimension r times the unstable counter-sets. OK. In the renormalization operator, it goes from H to H. The idea is that we take a point in this wall. It intersects. So there is a small piece of this stable set here. We take a backward iterate by the dynamics. Since the dynamics contracts in the stable leaves, a backward iterate expands. But we have this invariant foliation. So the leaf through this point will be sent to another leaf there. So the leaf through this piece, the intersection of the leaf with this piece will go to a leaf there. And this leaf corresponds to a new point in the wall which is the renormalized, the image by the renormalization. So each small piece of the construction of the stable counter-set gives rise to renormalization operators. There is a lot of possible renormalization operators, each corresponding to a cylinder, say, or to a piece of the construction of a counter-set. And a compact record set, well, is a compact set contained in the wall, such that for each point in the counter-set, there is at least one renormalization operator that goes to the interior of the counter-set. And we say that a function satisfies a compact record and criterion if there is a compact record and set. So the, well, this is an open condition. I have no time to explain. The theorem is that the compact record and criterion is sufficient to imply the presence of an air co-dimensional blender. So it implies that any manifold close to a leaf of the strong stable foliation, still unclose to a leaf, persistently intersects the unstable lamination of the rock shoe. This implies, in particular, that the image of stable counter-sets by differentiable maps contain open sets. And the main theorem after that is that, indeed, for most rock shoes with upper stable dimension larger than air, indeed, they have a recurrent compacts. They satisfy the recurrent compacts criterion. So they are air co-dimensional blenders. So we show that air co-dimensional blenders are typical among rock shoes. We'd have some chance of being at per-dimensional. We have enough dimension. So the stable counter-sets must have dimension larger than air. And if the upper stable dimension is larger than air, then typically the rock shoes are air-dimensional blenders. And well, in order to prove this, we use a version of Erdos probabilistic method that, unfortunately, I have no time left to explain. But the idea is to construct a family of perturbations with a very large number of parameters and to show that for most parameters, we create a recurrent compact set. We use a general version of master's theorem to construct a good candidate for a recurrent compact set. And we prove that this good candidate works for most small perturbations of this family of perturbations with a very large number of parameters. Well, very impressionistic description of the argument. But I tried to explain a little bit the geometry of this family of dimensions and the structures. So there are these two parts of the work. We restrict the study to rock shoes which have dominated the compositions in all dimensions, which are relevant to this work. And then we do something, in some sense, similar to this joint work with York Walls. And we should stand this joint, the first results with Zulu, with Walliston-Silva. And there are a lot of technical things that I have no time to explain. So thank you very much for your attention.