 So it's going to be a fantastic conference up to now and I think that it will continue for the second week too. So the talk will be about measures of high entropy. So it's kind of a bit beyond the measures of maximal entropy. So measures of high entropy for partially a class of partially hyperbolic dynamics. So fortunately Pablo and Kadim said about partial hyperbolic and you have seen also others examples. And you know what is measure of maximal entropy. So measure of maximal entropy is a measure which attains its entropy is equal to the entropy topological entropy of the dynamical systems. So I was really interested to understand how is how are the measures of high entropy. The measures whose entropy is close enough to measure of maximal to the topological entropy. And it was just a curiosity and after working something nice happened and we realized that well we realized because this result is a joint work with joint work with Jagang Yang. So it is a result down by Facebook in fact. We communicated too many times by Facebook. Okay. So let me go directly to what is the conjecture and what is the theory and what are the ideas of the proof. Two parameters or two notions are important to define a causticity of dynamics. One of them for me here to this talk is positive topological entropy. So if you have or metric entropy if you have a measure whose entropy is positive you say that this measure is caustic or another thing which is a criterion for causticity are the leopon of exponents of the measure. So you consider a measure and you consider it's leopon of exponents. Suppose that the measure is a ergodic mu. These are the leopon of exponents of the measure and if you have some positive leopon of exponents you say that your measure is is caustic. So you have kind of complexity on your dynamics. So you can ask if you have a dynamical system whose entropy topological entropy is positive what happens with the leopon of exponents of its measures of maximum entropy if they exist. So what about so let me just define something here a measure mu an invariant mu ergodic for the dynamical system f from m to m is called hyperbolic if all of the leopon of exponents of mu are different from zero. So you don't have any zero leopon of exponents. So this is a very good context in C1 plus alpha dynamical systems because you are in the context of passing theory you have stable manifolds unstable manifolds etc for typical points of the measure mu. Okay the very general question is that the question in fact is that how much how much hyperbolicity is seen by measures of high entropy. Okay for systems whose entropy is positive for systems for f with topological entropy positive. Okay in very general context you may you may not have hyperbolicity of even measure of maximum entropy for a system whose topological entropy is positive. So let me go directly to the to our context which is the context of partial hyperbolic partially hyperbolic dynamics with one-dimensional central central bundle. So you remember ES plus EC plus EU is the invariant tangent bundle the composition of the partial hyperbolic dynamics and I'm assuming that this bundle is one dimensional and moreover in here in in our context today we suppose that this central bundle is integrable and in fact it is integrable to a circle bundle so we have a central foliation foliation is a circle bundle. So let me give you an example to to simplify our life and go from this general thing. So I have this matrix 2111 and then I multiply it by identity. This is a partial hyperbolic dynamics a central bundle is by this is identified here and the stable and unstable are given by the anus of a stable and unstable bundle and then you you have a system which is exactly in our context. I will not I will not put all of the technical condition for the moment and you can perturb this dynamic if you perturb F to G you have if you perturbation is is a small enough in C1 topology then G also satisfies the also partially hyperbolic and it has one-dimensional one-dimensional center of center foliation again by compact leaves okay so the leaves central is persist. So in this class so in this class we would ask about the measures of maximum entropy so how are the measures of maximum entropy or even high entropy measures are? So there is a result we had done it by Federico and Hanna myself and Rau that says if if F is a partial hyperbolic dynamics in these conditions so you have let me just put it a special partial hyperbolic in manifold M it can be M-dimensional but we are always assuming that the central bundle is one-dimensional and suppose that you you you have central foliation which is which is given by circles a compact leaves you assume that you have integrability of course and some very an open and dense condition which is called accessibility I will just explain to you and accessibility then you have the following dichotomy then the following dichotomy for the measure of maximum entropy occurs either there exists the unique measure of maximum entropy measure of maximum entropy which is ergodic and the leapan of exponent the central leapan of exponent of mu is zero in this case so the hyperbolicity of measures for partial hyperbolic dynamics is just is just verifying the leapan of exponent on the central bundle because here you have positive and negative exponents so just the central bundle central leapan of exponent is important so here is the history is about one number which is the central leapan of exponent as it is one dimension so there is a dichotomy as I am writing here either your system has a unique measure of maximum entropy and in this case you have in this case you should have the measure should have zero leapan of exponent here in fact you you you mu and f is a Bernoulli system or there does not exist unique or there are strictly larger than one measures of maximum entropy ergodic measures of maximum entropy mu mu one and mu one negative mu and negative such that the leapan of exponents of these measures are positive and the leapan of exponents of that those measures are negative and you don't have any measures of zero leapan of exponent which is ergodic so lambda c of mu i plus are positive and on the c of mu i minus are negative and they are just finitely many ergodic measures of maximum entropy whenever I say measure of maximum entropy I refer ergodic measures of course you can combine these measures and get a measure with zero leapan of exponent so and in this case you have also some finitely extension of Bernoulli but here it is not I am not just interested in Bernoulli properties so I'm interested in this dichotomy either you have a unique measure of maximum maximum entropy or you don't have an in this case you shouldn't have zero leapan of exponents so the the hyperbolicity of the measure is the matter of uniqueness okay or the uniqueness is a matter of hyperbolicity of the measure of maximum entropy and another important thing is that this case is kind of non-generic case so so here in this case you should in this case f in fact f and mu is is a rotation extension it's a rotation type dynamics rotation type dynamics so you can you can have an example here a an us of times irrational rotation for instance you consider rotation it is an example of partial hyperbolic dynamics which satisfy the hypothesis and as it as you you consider rotation then your leapan of exponent is zero and in this case you have unique measure of maximum entropy and you are in the in the first case so if you if you consider the question of course for this case the answer is negative so if if you you have for instance here a measure which has maximum entropy and it is not hyperbolic so it is not reasonable to ask for measures of high entropy to be hyperbolic you have measure of maximum entropy which is not hyperbolic what I will say is that this is the unique case this is the unique obstruction so if your measures of maximum entropy are not hyperbolic are not sufficiently hyperbolic then you have kind of rigidity your dynamics should be this kind of dynamics it should be rotation type so let me now go to the announcement of the theorem precisely theorem that if f belongs to a special partial hyperbolic dynamics of M and if mu n is a sequence of measures such that the entropy of mu n converts to the topological entropy of f and if lambda c of mu n converts to zero then f is of rotation type so or equivalently or there exists epsilon and delta such that for any measure whose entropy is larger than topological entropy of f minus epsilon then the Lyapunov exponent of this measure should be necessarily larger than delta it says that all of the measures whose entropy is close enough to the topological entropy should have definite exponent nonzero exponent or if you have exponents go converging to zero then your system should be really rigid your system should be in rotation type but a rotation type a bit technically speaking says that there exists it is conjugate to a system that there exists an action of S1 to your manifold isometric action isometric action which just leaves invariant the central for the action leaves invariant central for the action coming to you with your dynamics so okay this is the the main result and well sometimes you you you look for something and you find another thing here appears that it was really a conjecture to understand high entropy measures and then to to prove this we realize that we can understand better by the techniques to prove this theorem some another known results which are called invariance principle and I will try to go inside and say you why why I think that we can understand better and they even have some some nice interpretation of this in so-called invariance principle okay well we just say you that there is some there there is some corollary in fact some our theorem I think it's shed light on some of the conjectures also there is a conjecture by Lorenzo Diaz and Gorodetsky they say that if for generic dynamics in fact for generic C1 generic dynamic in diff one of m either is hyperbolic or generic f either f is hyperbolic or or f has a measure of a non hyperbolic and non hyperbolic hyperbolic ergodic measure well with some variation of this conjecture also are written are claimed by them but what I'm saying here is that if you if you restrict yourself in this class if you consider this class of partial hyperbolic dynamics you cannot you cannot look for measures of ergodic measures which are not hyperbolic among the measures which have low entropy so so the place to to look for measures which are not hyperbolic is typically among the measures of low entropy so or in other word unless unless your system is very rigid your system is itself kind of isometry on the on the on the fibers in the partial hyperbolic case here then you can't expect measures which are not hyperbolic among the measures which are which have high entropy in fact in this class of systems in three-dimensional and robustly transitive systems Lorenzo Giaz and Christian Bonetti and Jair Oboki they can find they can in fact prove that either you are hyperbolic or you have heterodimensional cycle and then by heterodimensional cycle they can really show that if you are not hyperbolic then you have measures which are not hyperbolic and ergodic so you you construct really by hand these measures by by means of my by means of non hyperbolicity of the dynamics by means of heterodimensional cycles okay I cannot enter I don't need to enter to the details yes I say that I say that if it is not hyperbolic then you should look for measures which are not hyperbolic among the measures of low entropy you can't expect among the measures of high entropy okay so it's just kind of natural to expect of course but it's difficult to prove even you go to the context I did not say stated informally just I stated formally a theorem and what I expect is that I will I will formulate by a context a little bit later so let me put it the theorem here so let me go here and for instance make a conductor to answer your question so suppose that you have a dynamics which is partially hyperbolic let me be just one dimensional center dimension of EC is one and suppose that and suppose that all measures of maximal entropy ergodic measures are hyperbolic maybe I put it as a question is it true is it true that all high entropy high entropy measures are hyperbolic I expect yes my expect is my hope is to answer positively and as you are seeing I am I am answering positively in the context of circle bundle dynamics circle bundle central bundle and it is true also if you consider a partially hyperbolic dynamics which is homotopic to an Anasov for instance again it is true but it is very easy to see so the answer is yes if f is homotopic to Anasov and we are we are answering positively also if it is if the circle if the central foliation is circle by circle by compact leaves okay so let me now go to what we found then we wanted to prove it we can we can put it I didn't think maybe I didn't think on I think yes you you have measures with zero exponent that approximates your measure of maximal entropy yes I think if you if you permit higher dimensional center then you you should be able to do it I think that yet for the moment I have I have nothing but okay so now so okay I wrote down the the the context that the theorem that we had with Raoul and Hanna and Federico that says that in this context you have either you have isometric extension of Anasov system or you have a bunch of measures which are ergodic which have positive exponents and then some of them have negative exponent so the proof of this theorem relies on another one which is this theorem is immediate corollary of this theorem and says that theorem so maybe be so this is theorem a so again is the same so it says that if F if F belongs to partial hyperbolic a special partial hyperbolic dynamics that I just said you are and it is not rotation type so suppose that mu n is a sequence of ergodic measures ergodic invariant measures such that the entropy of mu n converts to the topological entropy of F then the apple of exponents central upon of exponents of all of mu n are non-positive then then mu n should be a combination of exactly this mu i minus up to mu l minus then mu mu n well mu n converging to mu then mu is exactly a convex combination of mu i 1 mu i mu 1 minus up to mu l minus this is comeback convex combination so at the first moment you can think that okay it should be trivial to prove that measures of high entropy should be hyperbolic because your measure of maximal entropy are hyperbolic so you have measures in you are in this case either you are rigid or you have proved previously that your measures of maximal entropy are hyperbolic so you know the exponent in the one-dimensional case in this one dimensional case the Lyapunov exponent of a measure is so just just dreams or non-proofs the Lyapunov exponent of mu is the logarithm of derivative of F along ec with respect to the mu so this is a continuous function and if you have a sequence of measures whose entropy are going to measure of maximal entropy so typically these guys should converge to measure if mu n converges to mu lambda c of mu n should converge to lambda c of mu and then you can say okay if these measures are non-hyperbolic if they have zero Lyapunov exponent they the accumulation measure the limit measure should be also non hyperbolic but of course the problem is that you do not know if the limit measure is ergodic so typically this can happen it happens you you have a sequence of measures which are ergodic and then they converge to a measure which is not ergodic and and no okay they have zero exponent and the limit measure also has zero exponent and you can have a combination of this measure with zero exponent okay so let me know maybe no I can shift to to another topic which is used to prove these results these theorems in fact so it is clear that theorem B immediately implies theorem A because if if you know that the high entropy measures the accumulation point of the high entropy measures with non positive exponent is always a combination of these measures these finitely many measures ergodic measures then what you can conclude is that never you can have very small Lyapunov exponent because because this measure any measure in this convex combination in here has a Lyapunov exponent which is bounded by the minimum of Lyapunov exponents of these measures so let me just before going to the proof yes okay maybe I this part is equivalent so theorem B implies immediately theorem A this is because suppose that suppose that you have a sequence of measures that are converging to mu and the Lyapunov exponent is also converging to zero you can assume that all of them are non negative so so the same result works for none non negative also so then you change here to mu i plus so suppose that you have a sequence of measures whose exponent goes to zero and look and lambda c of mu n all of them are non positive like in theorem B then what your mb says is that mu should be a convex combination of the measures which has negatively upon of exponent this finitely many number finitely many measures of negative exponent so of course mu also has a major exponent the exponent of mu should be should be larger than the minimum of the exponent of mu i minus so you can't go to zero okay so now we want to understand why a sequence of measures with negatively upon of exponent should converge really to a convex combination of these measures with negative exponent why no measure of positive exponent is invited to the limit point of mu n never a measure of positive exponent is invited in the limit of a sequence of measures who have negative exponent this is the point so here I would like to to remind something which is called invariance principle but it is it has many names in fact and it has roots the roots of these series of theorems I think I should put maybe first and bear led rapier and I will aviana I will not be I like Pablo I will not put men in but I think that really these results are in the building block of the invariance principle so what the invariance principle says is that is that if you have so let me go exactly exactly interpret the invariance principle in our context and not going to the general context so you have this dynamics which is partially hyperbolic it is it has a central affiliation you consider a quotient dynamic dynamics which is just quotient of your manifold by the central for the Asian so then here you have an analysis of the analysis of homeomorphism okay so here you have a partial hyperbolic dynamics here is central for the Asian then when you quotient down you have a dynamics which is an awesome hyperbolic and what the invariance principle say say essentially is that if you have a measure if you have mu if mu is a measure invariant by f which is your dynamic in this co-cycle so here you have a co-cycle if you have a measure which is invariant by the co-cycle and the Lyapunov exponent of mu is negative along the fibers then then the disintegration of the measure mu along the central for the Asian is you invariant by you invariance of this integration of the measure along this central for the Asian we see the means essentially that this this measures the measure mu along the central for the Asian is essentially invariant is invariant along the holonomy by the unstable foliation. So you have a holonomy here with the unstable foliation. And it says that when you go from here to here, you see exactly the measure which is here, the disintegration of conditional measure here. So this appeared for random products of matrix, for linear co-cycle. The idea here is fundamental. And here, Arthur and Marcelo, they generalize the linear co-cycle case to non-linear co-cycle case, which applies in the partial hyperbolic dynamic case, for instance, in our case. And what I am saying here is that we use some of these results, some basic results, previous results, to recover another proof of these results in this special case. So this is the kind of invariant principle. And let me just give some key points. Why do we really expect this thing? Why do we really expect that the measure should be a convex combination of negatively upon of exponent measures? Yes? No, no, non-positively upon of exponent. Yes? If it is non-positive, then you have the thing is that if you don't have any entropy along the central foliation, then your measure should be U invariant, in fact. The conditional measures are U invariant. OK, so let me just say you what we prove here before. Proposition is following. If mu is a measure, invariant measure, ergodic, not even invariant measure, and by F. And suppose that the entropy of the measure with respect to unstable foliation is equal to the entropy of the measure, then the measure mu is U invariant. So let me say you that it is a good interpretation of the invariant principle, at least for our purposes. And I think that it is also itself is a good interpretation without considering this application of invariant principle. Because it doesn't have any Lyapunov exponent in its formulation. So let me say you that this implies, for instance, the usual invariant principle. So if your Lyapunov exponent is less than or equal to 0, then let me say if the entropy of F with respect to the unstable foliation, I didn't define it, but it is defined as usual. You consider the unstable foliation. You consider a partition which is measurable, and it is adapted to your unstable foliation. And then you define the entropy of this partition. You need that this partition is increasing. This partition is generating. And then you define the entropy of unstable foliation as exactly the entropy of this partition. Maybe I arrived to define a bit precisely just in two minutes. So if this guy, if the Lyapunov exponent of F along the central foliation, central bundle, is less than or equal to 0, then the entropy of F, we know that by Le Drapier and Young, is equal to the unstable entropy plus the Lyapunov exponent here if it is positive, but we don't have any positive Lyapunov exponent along the center. So you don't have any contribution of the center to the entropy of your measure. So this is kind of the building block of our proof that says that if you don't have any Lyapunov exponent along the center, of course, you don't have any kind of entropy along the center. And we claim that for the invariance principle, you need just this. So if your Lyapunov exponent is non-positive, then of course you are in the hypothesis of our proposition. Then by the proposition, mu is you invariant also. So you recover again the invariance principle. So let me just finalize the idea of the proof of theorem A. In fact, theorem B, because theorem A is just an immediate corollary, idea of proof of theorem B. So you have a sequence of measures who are converging to mu. And we are assuming that the Lyapunov exponent are non-positive, lambda c of mu n are non-positive. So what happens is that the entropy of mu n with respect to the dynamics is equal to the entropy of unstable foliation. I put it plus nothing. This is equal. And what happens is that you are converging here. We are assuming that entropy of mu n is converging to the entropy of f, the topological entropy of f. And what happens is that these guys varize semi-continuously. Hopper semi-continuously. And if you consider the limit when n goes to infinity, you conclude that the entropy of f is less than topological entropy of mu. But on the other hand, you know that this guy should be less than the topological entropy. So you have the equality. And in fact, you have equality with respect to entropy of mu. So the entropy of mu is equal to the entropy of unstable foliation. If the entropy of mu is equal to the entropy of unstable foliation, then this proposition, which is kind of invariance principle, says you that your measure should be u invariant. It means that your measure, when you consider the conditional measures along the center foliation, should be invariant by the u-holonomy. So now I will draw just a picture. So you know here you have your measure, which is invariant by the u unstable-holonomy. And if a measure mu is u invariant, if mu is u invariant, of course, we don't know if mu is ergodic. This was the point. We don't know the measure mu is ergodic. We want to prove that it is a combination of negative central exponent measures, these measures. So if mu is u invariant, then all of its ergodic components are u invariant. Here I am skipping a bit because, well, it is written in the appropriate. But I think that it is good to see the u invariance of the measure, the conditional measures, as in the equivalent way, which says that if you have a measure mu here upstate, you project it down to a measure which is nu. This is kind of general. We don't need this to be topological entropy, maximal entropy measure, in fact. If you project the measure down here, then your measure here has conditional measure, nu u and nu s. So what does it mean that mu is u invariant? Mu is u invariant is equivalent to say that the projection p star of mu u, so always we talk about the disintegration along the central foliation, no change in my mind. And it disintegrates along the unstable foliation. So saying that the conditional measure along central foliation is invariant by u-holonomy is the same thing as to say that if you disintegrate along the u foliation, then projecting down it, you get exactly this nu u of p of x. So this is kind of just abstract measure theory that says you that if you want unstable invariant, then you go here and then your projection should be exactly nu u. And then this helps to say that all ergodic components are also invariant. In fact, you can prove it just by this definition also. But this will be useful in the final calculations. OK, so what we are, we are saying that our measure mu is u invariant, so all of its ergodic components are u invariant. No, you have a bunch of measures which are ergodic. You have a finitely many number of measures which are ergodic, mu i minus and mu i plus. And your ergodic components should be among these measures. So to finish, it is enough to observe that you can't have any measure in the ergodic component of your measure mu, which is also s invariant. So these measures are u invariant. The counterpart of them are s invariant. And the point is that no ergodic or if any ergodic component is s invariant or belongs to mu i positives, then you have rigidity or rotation extension, which we are avoiding. So if you have a measure which is u invariant and is s invariant, then by accessibility, you exactly like the previous result that I just removed here. If you have a measure which is both s invariant and u invariant, then you have your system should be rotation type. So isometric extension of an ASSOF. And OK, so we conclude that the invariant measures should be exactly u invariant or should be a convex combination of these guys. And it means that we prove theorem b, in fact, by this. And theorem a is a corollary. Just to finish in five minutes or maybe, yeah, five minutes, that just to highlight this proposition, so all we want to raise this, maybe this picture now. This picture is good. We have here a measure mu, which is kind of that limit measure, for instance. You project down this measure, p star of mu. You get a measure mu here, which has the same entropy because you have one dimensional here. For instance, again, it's used. And you have u invariant means that p star of mu u, this integration along unstable fallation, is exactly this integration of the measure mu, mu s. So you come down here. Your measure here should come down here. It's mu u of p x. So this is the u invariant. And the proposition says that if the entropy along the unstable fallation is the same as the entropy of your original system with respect to the measure, then you should have this property. You should be u invariant. Well, I should say you that this is a calculation of one page calculation, just nothing more. And it uses, essentially, the Jensen inequality and the existence of Markov partition for this dynamic here. So what I want to emphasize is that we recover the invariant's principle. I don't say even more than invariant principle because invariant principle has many versions and many, many other things. But at least in the case where you have a base which has a measure with product structure, which is the case of results, some of the results, if you have a measure whose entropy is equal to the entropy along the unstable fallation, which is the case when you have zero exponent or negative exponent, non-positive exponent, then what we are saying is that by means of the Markov partition here, you can really prove that your conditional measures here, the conditional measures along unstable fallation, should be the same. I mean, this one. All of them, this is the projection. So if you take any point, any unstable plaque that goes to the same unstable plaque of the onosophomia morphism, you should see the same measure. So this means that you have u invariance. And this is just a trick of Jensen type inequality, which appears typically in this kind of literature. And you prove the male result. OK, thank you very much.