 Thanks, Stefano. That's a great pleasure to me to speak here again, and especially because in the audience we have people that finish PhD in Maceo after a period. So today I will speak about a problem that I've been working for a while. And I would like to talk, I divide the talk in three parts. The first part is a main question that I want to discuss. The second is an example where we can get uniqueness of equilibrium measure for some partially hyperbolic host shoes. And finally I would like to discuss about maximum entropy measures for random local details. So let's begin with the main question that I would like to discuss. So let's assume that we have a map, a C1 map from a manifold. And given a point, let's denote by lambda 1, lambda 2, lambda d. The Lyapunov exponents at this point, if they exist, they don't need to exist. So if they exist, I will write this. So let's denote by ek, the set of points, such that the Lyapunov exponents exist at this point. And we have exactly k negative exponents and d minus k positive exponents. So ek is an invariant set, maybe it's an empty set, but I don't care. And maybe probably it's a non-compact set. So the main question that I would like to discuss is, can we prove that under some assumptions on the map f, the map f restrict to the set ek has at most one equilibrium state for any hold the potential fight. So I highlight the word restricted because I would like just to consider measures that see the set ek. So that's the question that I would like to start. So ek is the set of points with k negative and d minus k positive exponents. Actually, we can split the manifold. We can decompose the manifold as the set is zero. So the set of points with positive exponents only, e1 up to ed. So this is the number of negative exponents. Yes, this index is the number of negative exponents. So here we have the set of points with only negative exponents. And here, let's see, we have the rest. The set of points with some zero exponents or the set of points that do not have exponents. So each of these sets are invariant. I would like to discuss the map restricted to each one. So the conjecture is, if you assume something reasonable, I will explain what I mean by reasonable. Then any hold the potential has a unique equilibrium state on the set, restrict to the set among the measures that see the set. Again, the set can be empty or the measures that are invariant maybe don't have an invariant measure on the set. So by mild condition, I mean that condition about mixing properties of f on the set ek and conditions about the critical points if they exist, or the splitings, the Oseladette splitting. More precisely, for instance, a problem, a more precisely problem. Let's assume that you have a topologically mixing map such that the Oseladette splitings dominate on the set ek. Can you get uniqueness of equilibrium state for hold the potentials on the set ek? Or in a similar way, let's assume topologically mixing for local difference only. And let's consider, for instance, the situation of E0. Can we get that any hold the potential has a unique measure, unique maximizing measure, unique equilibrium state restrict to E0. So by restrict, again, I mean the measure that maximizes this, the entropy plus the integral, over the measures that give full measure to the set ek. So we get consequences if you can prove this. So maybe you can get real unique equilibrium states, not only restrict the equilibrium state, but a real unique equilibrium state in some situations. When we can get that the set ek has some, plays some role in the pressure. For instance, maybe if you can get uniqueness on the set ek, and we can, if you assume that the potential satisfied something like this, the pressure of the complement of ek is smaller than the pressure of ek, we can get uniqueness of equilibrium state. And another thing that we may get, somehow related with the talk of Felix and the talk of Juan, is that we have phase transitions when the measure decide to jump from one set ek to another set ek, from the set em to the set en. So maybe we have finite number of phase transitions. So that would be possible consequence for this. So we have now a lot of work that I would like to mention quickly, because the list is long, and most of them are here in the conference. So at symbolic level, we have breakthrough results by Sariq and Bouzi, and connecting maps with symbolic space. We have several works of Bruin, Pinheiro, Pezin, Saint-Zang, and we also have a different approach by Klemenhagen-Thompson via specification. So it's a different kind of specification, a weak specification property that one can use to get uniqueness of equilibrium states. And several examples from many people. Maybe an interesting example that I just realized, just read the paper, it's the example of Tazebi Rodrigues hats, where they get some examples of measures of maps, of partial hyperbolic maps on dimension three with two equilibrium states, but they have different signs. The entropy, the aponov exponent in the central direction, has one measure has positive sign, and the other measure has negative sign, and the map is topological mixing. For one-dimensional maps, we have several works. At dimension two, we have a recent theorem by Sariq that proves something like I stated before, and at dimension three, for flows, we have even more recent work by Lima and Ledrapie and Sariq. All of these examples and work are pretty much in the setting that I described before. I don't know a situation that are not in that kind of setting so far. I would like to know. I've seen people but so far I have not found any indication that that thing would not be true. I think that's true. So today I'm going to talk about two situations. The first situation is a partial hyperbolic host shoe that we studied with some co-authors, and the second situation is random maps. So let me start to describe what is the partial hyperbolic host shoe that I would like to discuss today. So maybe I'm going to jump this. I go to the picture. It's more... then I go back. So how we define the map? We have a cube, okay? And in the cube I will consider the direction x as a uniform contracting direction. The direction z is a uniform expanding direction, and the direction y is a central direction, and the central map is defined by this map, okay? So near zero you have expansion, near one you have contraction. So essentially the map does something like this. It contracts in the direction x, expands in the direction y, and then it folds in such a way that this side here, this pink side, enters in the cube again with face in the side of the cube, okay? So the formal definition is like this. So we define it on the... we divide the cube in three parts. So let's say r0 and r1. So the important part is r0, the part below the cube and r1 is the part above. So we fix constant, the expansion contraction constant. And we define, as I told you before, in the x-directional contraction, the z-directional expansion, and in the central direction, the map f. And the map f is the time one of our vector fields. So in the upper part of the cube, we have a hyperbolic map, okay? So we have contraction, expansion, and expansion again. But here, actually here's contraction. But here I do a flip in such a way that this map is pictured here. So what are the properties of this map? This map was introduced by Diaz and Orita, Sambarin and Hughes, and the map has some properties that you can check. The first property is that the point q, 0, 0, 0, is a fixed saddle with index two, and the point 0, 1, 0 is a fixed saddle with index one. So I'm going to show another picture very soon. So they prove that the map has a third-dimensional cycles, so the hyperbolic saddles q and p have different index, and they intersect. The stable manifold of p intersect the unstable manifold of q, and the stable manifold of q intersect the unstable manifold of p. The homo-clinic class of q is contained, it's trivial, it's q, and contained in the homo-clinic class of p, and the homo-clinic class of p is the maximum invariant set, and it's no wonder set of the map f. Okay, what else? It's possible to semi-conjugate this map with a horseshoe, and this horseshoe here is the horseshoe that do not allow the transition 1, 1. So we have a map that send a point in the cube in a sequence of 0's 1's, and in this sequence we do not have 1, 1. So this is a semi-conjugacy. And here is a very interesting feature of the horseshoe. So the map pi, the semi-conjugacy is not 1, 1. So we have some points that have a segment as a central manifold on the no wonder set. And this set corresponds, for instance, the pre-image of a cylinder like this is a collection of points that belongs to the same central manifold. So the homo-clinic class of p has infinitely many central curves that are not points. So this is something that they call now porcupine horseshoes. They have several spikes on the horseshoe and the spikes project by this map, by this projection in a single point in the shift, in the shift sigma 1, 1. So this is an interesting dynamic. Here is a picture of the heteroclinic test section that appears. The unstable manifold of p intersects the stable manifold of q. So in particular, if you take a point nearby here, the point eventually will follow this trajectory and will approach q again and will eventually return nearby here. So one problem that you may ask is what you can say about the invariant measures for these maps? We studied the invariant measures of this map, of this kind of horseshoe. And with Hennel Leplac and Isabelle Hughes, we obtained some results. The first result is the following. Given any continuous function, you may prove that that exists equilibrium states for this horseshoe. And you can prove also that there is a residual set of continuous function that the equilibrium state is unique. Pretty much because we are able to prove that the entropy is up as a continuous in this situation. One more interesting result that we have obtained is that in this kind of horseshoe we have phase transitions. So there exists a number such that if you consider the family, phi t, this family here, t times the logarithm of the derivative in the central direction and you study the equilibrium states of this family here, we were able to prove that there exists a number t0 and we can describe this number. So the number is the supremum of the entropy over the one minus the central, centrally upon of exponents where we take the supremum over all measures that are not equal to delta q. And this family has a phase transition meaning that for t bigger than t0, delta q is the unique equilibrium measure. For t smaller than t0, any equilibrium measure has negative central exponent. And for t equal t0, there are at least two equilibrium measures. One with positive exponent and another with negative central exponent. Actually the equilibrium measure with positive exponent is the delta q. So if you make a picture, maybe you will recognize this picture. We saw it before. The picture is something like this, a picture of the pressure. 2 phi t0. So here we have the point t0 where we get the phase transition. And the question that we pose to ourselves, so we can prove that we have uniqueness here, unique equilibrium measure. We can prove that all equilibrium measures that are for t in the left of t0 have the same sign of the exponent that have only negative exponents. So in particular they live on the set e2. Here the equilibrium measure would live in the set e1 and here they live in the set e2. But we were not able to show that they are unique in the left of t0. So that's the goal of the talk today to discuss a result in the direction of the uniqueness of the equilibrium state for potentials on this whole show. So the question is what conditions you can ask about the potential to guarantee uniqueness of the equilibrium measure? So there are some results that were obtained after we published the previous theorem. And the first result is that if you assume that the function is constant on central leaves then you have uniqueness. And also more recently, if you assume that the potential is constant on stable manifolds, in the case in the direction x, then you can show uniqueness as well. But so the approach that the second result of real cicada, the approach that they use is to consider a restriction of the map to center unstable manifolds and to show that there is a kind of Perron-Frabenius theorem for the map defined on levels of this whole show. So today I would like to talk about uniqueness of these, for all the potentials in general for this whole show. So we obtain uniqueness and we obtain uniqueness assuming the following conditions. So let's assume that the potential is holder and the pressure, the supreme of the entropy plus the integral over the set of measures that have only negative exponent is bigger than the supreme of phi. Then phi has a unique equilibrium measure. So that's the condition that we need about the potential phi to obtain uniqueness. Just a remark, the set of measures, in this example, the set of measures that have negative central exponent is every measure, every gothic measure minus the measure delta q. So we described this set in the before. So in particular I'm taking the supreme over all measures and accept delta q. And one situation where you can check this is when you have a potential such that the maximum minus minimum is more than topological entropy of this measure, 1 plus square root of 5 over 2. So let's now talk about different example, the same setting of same kind of question that I posed in the beginning of the talk. But the example is about random maps. The result is about random maps. So I'd like to start with the classical result about expanding maps. So for expanding map, we have that the topological entropy is equal to the logarithm of the number of pre-images. If you have a expanding map on a compact, connected Riemann manifold, you can show that the topological entropy is equal to the logarithm of p. p is the number of pre-images of any point. And you can show that there exists a unique maximum entropy measure. So that's the old result by Bohm. So a natural question would be to consider local defile morphism. Given a C1 local defile morphism, you may define the topological degree as the same, the number of pre-images of any point. And so we prove it under some non-uniform spanning conditions about the derivative. That exists a unique maximum entropy measure. So that is also a related result by Geron Bouzier. And let me explain what is the assumption about the map. The map, the local defile morphism is not a spanning map anymore, but we need that kind of expansion to get uniqueness of a maximum entropy measure and to get a formula for the topological entropy. So the underlying assumption here is that essentially that measures with big entropy have only positive level of exponents. So in this situation, the condition that I would describe in a second gives to me that any measure with big entropy will see only the set of points with positive exponents. So the set of points is zero. So what is the condition? The condition that I assume about the local defile morphism is written in terms of lambda q of the derivative, the exterior product of the derivative at any point where I assume that this inequality holds and k is smaller than the dimension. So pretty much I take the biggest rate of expansion for the derivative, for the k exterior product of the derivative among the manifold. And I consider k smaller than the dimension, so I take the maximum over all of these rates and I assume that this is smaller than the logarithm of p. So if this condition holds, so the map is not expanding anymore, but it has some expansion, and some expansion means this, and this implied this first assumption here that any measure with big entropy will have only positive level of exponents. So if I assume this, we can get that the topological entropy is equal to the logarithm of p, and we can get the uniqueness of the maximum entropy measure under mixing assumptions. Okay, so what's the problem that I would like to talk today? It was a result with Rafael Alvarez. So I would like to talk about the random compositions of local defiles. So what we can say about random compositions of local defiles. So what I mean by random compositions? I mean that we have an invertible map, theta, preserving a gothic invariant measure p, and I will take a family of c1 local defile morphism for each element of x, I will take a local defile morphism f of x, and with these two ingredients I construct a skew product like this. f at a point x, y is theta of x applied to f of x at y. So if you look at this and you start to iterate f, you can write for simplicity fn of x, y as theta n of x, and this component here gets more complicated. So fn of x at y is this expression. So it's what we call a random composition of the maps. So for each of these maps f of x we have a topological degree p of x, and the following diagram is commutative. So we have a projection, a natural projection from the skew product, from the product x times m on x. So I consider the set of measures, set of all invariant measures, such that the projection of the measure on the direction x, so this measure here is equal to p. I will consider only the set of measures, the set of measures that have marginal p, and we consider the set of measures among these that are invariant. So in the talk I just will talk about measures that when you project on x, you get p. p is fixed from the beginning. So everything that I'm going to talk will be relative to p. So let's discuss an example just to illustrate what I'm talking about. Take two local diffuses, f0 and f1, and let's consider the p0, the number of pre-image of x0, and p1, the number of pre-image of f1. Let's consider a particular measure p as the Bernoulli measure on the shift, such that the measure of the cylinder 1 is equal to alpha. So then we have an iterated function system with a Bernoulli measure on x, and we can ask about the maximum entropy measure of this system. But again, when I mean maximum entropy measure, I mean maximum entropy measure that has marginal p. So that we call a relative maximum entropy measure. So what we can prove with the result that I'm going to show in a moment. We can prove that if you take a map f1 as before, as in the result with Marcelo Viana that I mentioned before. So if you take a non-uniform like spending map in the particular sense that I discussed before, and if you take alpha close to 1 and you take any local d field p0, if you take alpha close to 1, then the topological entropy of f is equal to this average. 1 minus alpha log of p0 plus alpha log of p1. So you can translate this as if you take a non-uniformly expanding map in this sense, then you take non-uniformly expanding local d field in this sense and you take any other local d field if you start to iterate them randomly with respect to the Bernoulli measure. If the Bernoulli measure concentrates too much on the non-uniform map, the result will be non-uniform. The random map will be non-uniform. And you can get the entropy formula and the maximum entropy measure relative to the measure p. So that's an example just to illustrate what you're going to follow. So if we assume that the map f is exact, topologically exact, then we have that that exists a unique maximum entropy measure relative to the measure p. Okay. So here is the definition of what I mean by what is the relative metric entropy. So it's pretty much the same thing of the entropy. But now we're going to consider partitions that refine the partition by points in X. So the partition p is finer than this partition. So in particular, it's a partition, it's a partition on the fibers as well. So we can define the entropy as this, limit of this. And by this number here, this number here is defined as the integral over X of the entropy of the partition pn of X, the usual refining partition. And maybe I would explain what is it. This is the disintegration of the measure mu. Given a measure, an invariant measure, we can disintegrate it on the fibers using Hawke's integration theorem. And we get this system of measures. And to compute the entropy of the measure mu with respect to the partition p relative to the map theta, I need to average the entropy of this system of measures with respect to the measure p. Okay. So I will skip this. Maybe quickly mention what is the kind of functions that we consider. We consider functions, phi, that are continuous on the direction y, on the fibers. If you restrict the fibers, they are continuous. And if you look at the direction x, they are integrable with respect to p. So that's what we call L1 of X with respect to X, the collection of functions that are c1 with respect to X and continuous with respect to y. So under these conditions, you define the topological pressure as the usual. But again, you need to average with respect to p. And you may define the transfer operator as usual, but now you have a family of transfer operators. So for each x, you have a different transfer operator and the transfer operator depends on x. And you may define tools for these operators by the usual relation. So the eigenmeasure, we call eigenmeasure system, the maximum eigenmeasure system is defined as a system of measures such that this equality holds. A system of measures such that when you apply the dual to the measure mu of that ix, you get p of x, mu of x. This is exactly what you expect when you compare with the expanding case, the expanding situation. Okay. We assume that first assumption about the set of local differs. We assume that the logarithm of the derivative of the supreme of the derivative is integrable with respect to any measure. In particular, this is true when this is bounded. The example that I gave before. So that's the first assumption. And these assumptions are necessary for the, for instance, for the azela, that's theorem for the Brinkatox theorem and all other results that we have in erotic theory. So these assumptions are pretty much common to get the usual results in the deterministic setting. And here we have the similar that we mentioned before. For each number k, for each number k, I will define the lim soup of one over n logarithm of the stereo product of the derivative at y. I'll define this as ck. So this rate of expansion of k volume at the fx. And then we define ck of x as the maximum among y. And the assumption that we require is that the integral of the logarithm of px, so this is the average degree, logarithm of the degree, is bigger than this number. Okay? And also we need a kind of uniform continuity of the derivative at p almost every point. So this is a kind of uniform continuity of the derivative. Under these conditions, we can get the following. Under the condition 2, 3 and 4, we can get that the topological entropy of f relative to theta, so relative to the measure p, is equal to the integral of the logarithm of px with respect to p. And if mu is a measure that's a eigenmeasure for the transfer operator, so it's a maximal eigenmeasure system, then mu is a maximal entropy measure. Moreover, if you assume that the map f is topologically exact, then, and the degree of the map is bounded, the supreme of the degree, so this is p of x that appear above, then there exists a unique maximal entropy measure, and this measure is positive on open sets. Okay? So that's what we got. Here I check how we can get the condition 2. So remember that the main condition that we need to assume is this condition here, 3. So we need to show this condition if you want to apply the result. There are other conditions, but this is the most difficult to check. And if you want to check this in the example, if you want to check this in the example, you just notice that this is bounded, this number is bounded. You have only two maps, so this number here is L, this is a local diff, and then you may take the maximum among k of these two numbers, and then when you want to compute this, you can split in two parts. The collection of points that have a lot of ones, so remember the example. The example is we have two maps, and I have a Bernoulli measure that give a lot of measure to the cylinder 1. So when we iterate randomly F0 and F1, I see a lot of ones. I see a lot of functions F1 appearing at a rate alpha. So I may define the set of points that see, that the sequence of the sequences that have a lot of ones appearing up to the moment n. So by a lot, I mean at least a rate alpha tilde, and if I take alpha tilde smaller than alpha, I know that this set here, when n goes to infinity, the measure of this set will converge to one. And then if I want to compute this, I split in the set an, and in the complement of the set an. So then I can get that here, you get this estimate, and below we get this estimate. This part here goes to zero, because this measure here is going to zero when I divide by n, so I divide by n, everything. So this part here goes to zero, and then we get this kind of a bound for this number. If I make alpha tilde go into alpha, then I get this estimate here, that this number is smaller than this. And finally, if I take alpha going to one, I may get this going to zero, and I know that this is smaller than the logarithm of P1. Then we get inequality that we want, okay? I run out of time. Maybe I'm going to show, I jump to the strategy of the proof of the theorem one about the host show. So what do we do in theorem to prove theorem one? Essentially, we record the host show in an infinite shift. So remember the result about the host show, that the host show is naturally coded in a finite shift, but the code is not one-to-one. So using this structure, we record in an infinite shift, and the infinite shift is, the record is one-to-one. But we need to throw something away, and we show that what we throw is, has not equilibrium measure, if we assume the condition about the potential. So in other words, we check the condition under the condition about the potential, the scheme that we get is hyperbolic in this scheme, and we get uniqueness for this scheme. I have also this scheme for random local defaults, but I'm going to stop here. Thank you.