 Thank you. Can you hear me? Yes, okay. So first of all, I would like to thank the organizers for the invitation to speak here. It is always a pleasure to be at ICTP. And second of all, I would like to apologize the experts in the room because my talks they will be fully dedicated to the students. So if you understand why symbolic dynamics are useful for dynamics on Thursday, I will already be happy, all right? You can stop me anytime you want to ask questions. So what I would like to start with is with an example. And this example was already mentioned by Federico last week. So what is the cat map? The cat map is just a very, it appears to be a very simple map on the two torus. It is, you get this two by two matrix which has integer entries and determinant equal to one. So you can induce a map on the two torus. And how is the action of this map? Well, that's what it does with the cat. So it stretches the cat in one direction. It contracts in another direction. And because we are considering the map on the two torus, you have to reassemble this fundamental domain to the basic fundamental domain that you are working with. So you get this piece here and you translate one unit to the left. You get this, you translate one unit down and one unit to the left. And you do the same thing with this thing here. So the cat gets distorted. My goal is to try to understand better dynamically this cat map. And how do we do that? Well, that's where symbolic dynamics is going to come. As you know, this map, it is a hyperbolic torus automorphism. So it has two eigenvalues. One of the eigenvalues has absolute value bigger than one. The other eigenvalue has absolute value smaller than one. So in particular, you know that two points, if they are distinct, they cannot be close together forever in the future and in the past. You have what is known as expansiveness, which means that if you get two different points, well, there will be a future iterate or a negative iterate or both. That will make these two points at least absolute apart. In other words, you can say that this map is sensitive to initial conditions. And it is exactly this property of sensitivity to initial conditions that is going to allow me to construct symbolic dynamics for this map. What is the idea, underlying idea of symbolic dynamics is that instead of describing what is the orbit of a point, so instead of describing what is the orbit of a point, what I'm going to do is that I'm going to discretize my space and I'm going to tell to each rectangle, to each all elements of this discretization, my orbit is going to belong to. So I'm going to discretize the space and I'm going to describe the orbit of x, not by saying exactly where x, f of x, f minus one of x and so on are, but by saying to which rectangle of this discretization, the iterate of my orbit belongs to. That's the idea. So when you consider the cat map, well, you can consider the cat map with respect to the x and the y axis. Why? Because we are used to the x and the y axis. And this is just the same picture that is in the previous slide. So what the cat map does with the fundamental domain of the torus, well, it stretches, so it becomes this parallelogram and when you reassemble, it becomes this thing here. So the idea is that x and y axis are not the right system of coordinates to look at when you consider the cat map. You have a much better system of coordinates, which is the system of coordinates given by the dynamics itself. And why do I mean by that? Well, instead of considering the x and y axis, why not considering the eigen directions? I'm going to look at what the map does in the eigen directions. So this is the big idea of Adler and Weis and of Sinai, which is to do the following. Okay, if I have this sort of hyperbolic torus automorphisms, well, why not considering these dynamically relevant eigen directions? In other words, what can I do is I will tessellate the plane with rectangles or with a fundamental domain like this, which is made of rectangles, such that each rectangle or each side of the tessellation is parallel to one of the eigen directions. So in this case here, you have these ones, which are parallel to the expanding eigen directions. And the other ones are parallel to the contracting eigen directions. Okay. And why is it better to consider this system of coordinates? Well, because the image of rectangles will again be a rectangle. So it has a much simpler geometry. And then it will be much simpler to understand what is the dynamics of this rectangle under the cat map. Actually, if you consider this fundamental domain, and you divide it into two pieces, pi one and pi two, what does a do with these two pieces? Well, it gets pi one and it stretches in this rectangle here. So when you reassemble it back to the fundamental domain that you are considering it, what it does is to cross pi two ones, and it crosses pi two pi one twice, right? This is this yellow part here when you put this down one unit and when you get this and you put one unit down and one unit to the left. And it does similar things to pi two, it stretches and the image of pi two becomes this very thin rectangle, which is cutting pi two ones and it's cutting pi one twice. All right. So the great idea of Adler-Weiss-Sinai is exactly, okay, let's change the system of coordinates and let's look at the system of coordinates that's relevant for the dynamics. Okay. So what is the main property of these rectangles that I'm considering here? Well, as I told you, I want to, instead of describing the orbit of a point, I want to describe to which rectangles it belongs to. So I would be happy if these rectangles have what is called the Markov property. And what is the Markov property? It's a property that tells you which ways the rectangles can cross. So if you have a rectangle R for which when you iterate it, it intersects a rectangle S, it has to intersect all the way down from one side to the other. So this is one allowed intersection. And this is a not allowed intersection. Why? Because the image of the rectangle intersects the left hand side of S, but it does not go all the way to the right hand side. Okay. And if you think why you want this property, it is because if you have some, some possible passage from one rectangle to the other, it means that you have a point X that belongs to this rectangle and hits the next rectangle. Well, now if you have another passage from one rectangle to another, it means that you have a point Y for which F of Y belongs to S and F two of Y belongs to T. And what you would like to do is to concatenate these two edges and get a point that belongs to R whose image belongs to S whose second image belongs to T. And if you do not have this crossing property, this might not be possible. For example, if the image of R is like this, and if the image of S intersects T like this, what is the, what is the second iterate of R? Well, the second iterate of R will intersect S here, and the image of this piece here, here will be this small piece here. So there will be no point that belongs to R whose image belongs to S and whose second image belongs to T. So our goal is to have this sort of crossing property, and in the case of hyperbolic torque automorphisms, just like the cat map, we can do that. So what is the final result for the cat map is that we are able to get what I call here a symbolic model for the cat map. What does that mean? It means that I got my space T2 and I cut it, I discretized it in finitely many rectangles. And this I'm going to call the set of vertices of my symbolic model. Well, if I have vertices, I also want to have edges. What are the edges in this symbolic model that I'm constructing are exactly the allowed passages from one rectangle to the other. So whenever the image of a rectangle intersects the other, you draw an edge from the vertex R to the vertex S. Okay? And then you got a symbolic space because what you have here VE is a oriented graph. In an oriented graph you can consider all the paths on this graph, which you can index by Z. So the sigma here is a symbolic space given by the Z indexed paths on the graph. And in this space you have a natural dynamics which is the action of the left shift. Okay? So sigma, sigma is what I will call a symbolic space which I will later call a topological Markov shift. So to get a symbolic model, I have to define a symbolic space which I already defined, but I have also to define a coding. And how do I associate a point here in the symbolic space to a point in the two torus? Well, it is what this map pi does and it is exactly defined by the property that I want to see here. I want to get a sequence of rectangles and I want to generate a point that's following the itinerary given by the rectangles. So that is only one choice for me to define this map pi, which is this infinite intersection here. Right? So if by any chance this infinite intersection here consists, is non-empty and consists of a single point, then I'm in good shape. And why is this the case? Well, it follows exactly from the dynamical properties. The Markov property allows me to concatenate finitely many rectangles and it actually allows me to prove that this intersection is non-empty. And the expansion and contracting properties of this hyperbolic torus automorphisms will guarantee to me that this intersection can be at most one point. Why? Because horizontally it will be exponentially small if you go up to time n. So if you go up to infinity, it will be, it will have diameter 0 horizontally and the same thing vertically. So it consists at most of a single point. So by these two properties here, I get my coding pi. Okay? And why is that good? Well, because I can conclude that, and you should regard this as f here, the cat map and the left shift are basically the same thing. What do I mean? I mean that I constructed a symbolic space with the shift and a coding pi that makes this diagram to commute. And also almost every point here has exactly one pre-image on Sigma. So this map pi here is one-to-one in a residual set of T2. So for the purpose of dynamics, we are in very good shape because we're able to translate the action of f in the manifold in the two torus by the very simple action which is just left shift in the symbolic space. And why is that good? Well, that's good because for example, as I said, you can see, you can iterate the map in a very simple way and also because you can count the periodic points that this map has. How many periodic points this map has? Well, it has at least as much as Sigma has because every periodic point here projects to a periodic point on the basis. Right? So it is good for iteration, it is good for counting periodic orbits and it is also good for constructing and analyzing the ergodic properties of the invariant measures. Right? So this is the goal. The goal of symbolic dynamics is to come up with a simpler system which is usually given by a symbolic model that allows you to get properties, ergodic and dynamical properties of your complicated system that you start with. Okay, so what are the models that we are aiming to construct? Well, this is just an abstraction of the model that I described previously. So in general, we are given a diffeomorphism and we want to find an oriented graph, G, given by a set of vertices that I will always assume that it is countable and a set of edges. And whenever you have this graph, you can consider the symbolic space generated by it and it's just the set of z-index paths on G and you have the dynamics acting which is the dynamics of the left shift. So given a diffeomorphism, we are aiming to find such sigma, such pi, which I will call from now on a topological mark of shift for short TMS. Okay? And it's not only sigma and sigma what we are aiming to get, we are also aiming to get a coding. So for each z-index path on this abstract graph, we would like to get a point in the manifold that makes the dynamics of the diffeomorphism that I consider to intertwine with the dynamics of the left shift above, such that I can reduce the analysis of properties below by the analysis of properties on this topological mark of shift. Okay? So this triple sigma, sigma pi is what I'll call a symbolic model for diffeomorphism and it is exactly what you are aiming to construct. Okay? So in the lecture, I also expect to talk a little bit about symbolic models for flows. And what is the symbolic model for flows? Well, when you analyze flows, what you do usually is that you consider a reference section and you analyze what happens with the first return map to the section. So the action on the flow, it's just a translation, just a linear translation. So what you are aiming to analyze really is to get some property with respect to the first return map. And in this first return map, if you have hyperboleicity for the flow, the first return map will be some sort of hyperbolic diffeomorphism. So for the return map, what we expect to see is exactly a topological mark of shift. You expect to model your system with in the return map with respect to topological mark of shift. So a symbolic model for a flow will be, first of all, we are given a topological mark of shift and we are giving a positive function defined on this topological mark of shift. It will describe to me exactly the time it takes for me to start in the section and return to the section. It is a root function. So whenever you have these data, sigma, sigma and r, you can consider the suspension space given by the stripper. What is it? It is the set of pairs vt, where v is in the symbolic space and t is below the graph of the function r and you have this identification here. Whenever you hit the graph of r, you come back to the section by applying the left shift. So this is the suspension space. It is a topological space and in this space you can consider the unit speed vertical flow. You start with a point in the basis and I will draw a picture to you soon to understand it better what it is. You start with a point in the basis and you start flowing at unit speed vertically above until you hit the graph of the function and what happens when you hit the graph of the function? Well, you remember the identification that you defined and then you come back to the basis section and then you continue flowing at unit speed vertically. So this is the unit speed vertical flow and this pair, the suspension space and this unit speed vertical flow is what I will call for now on a topological Markov flow for short TMS. Okay, so we already got the symbolic space or the symbolic, yeah the symbolic space and the symbolic dynamics that we aim to construct for flows but we also need a coding. So we are going to have this and a coding that for each point on the suspension space you associate a point in the manifold and this is exactly what I'm aiming to construct for flows. It is a triple sigma r sigma r pi r such that this first two coordinates defined to metatropological Markov flow, this third one is a coding that intertwines again the dynamics of the flow below with the dynamics of this unit speed vertical flow in this suspension space. Okay, so here is the picture again. All right? Questions so far? Okay, no questions? Okay, so before stating the results that the most recent results I would like to say a little bit about the old results and the old results are always for most of the time for the systems which are uniformly hyperbolic, either a nozzle of systems or axiomase systems. Okay, so just like I told you with that cat map example the first appearance for these systems, the first construction of symbolic dynamics for these systems was given by Adler and Weiss in 1967 which is exactly the two-dimensional hyperbolic total automorphisms just like the cat map and at the same time CNI also gave a construction that works for a much broader class of systems which is that of a nozzle of defilmorphisms. So these two were more or less at the same time and a little bit after Bohlen was able to go further in the methods of CNI which the methods of CNI if you want to know they are known as the methods of successive approximations. So Bohlen was able to extend this method of successive approximations to axiomade defilmorphisms. Okay, and these are the main results for hyperbolic, uniformed hyperbolic defilmorphisms. What happens in the case of uniformed hyperbolic flows? Well, Ratner was able also using the method of successive approximations to get this construction for a nozzle flows and Bohlen again in the same year was able to do it for a more general class of flows which is that of axiomade flows as long as you assume that the non wandering set has no fixed points. So if the axiomade flow is such that it's no wandering set has no fixed points, well you can code it, you can get a symbolic model by a topological Markov flow. All right, good. So what are the examples that these classes of systems include? Well, in the case of defilmorphism they include all hyperbolic total automorphisms and in the case of flows they include all geodesic flows on manifolds on which you have negative section of curvature. Okay, these are known to be actually a nozzle flows so by the result of Ratner you are able to get a symbolic coding for these guys. All right, so what are the main applications that you can get from symbolic dynamics and here is where I want to convince yourself that symbolic dynamics can give interesting applications for the dynamics of the system. So let me start with the case of defilmorphism and let me assume that you have an axiomade defilmorphism that is transitive and has positive topological entropy. Okay, so by getting the symbolic dynamics, by constructing the symbolic model, one first application is that you are able to count periodic points more accurately. So that would be a P which basically takes care of the possible period of your axiomade. If you assume the axiomade to be topologically mixed in this P will be equal to one but in general it's bigger than one and what you have is that along the multiples of P the number of periodic points is exponential and the rate of growth is given by the topological entropy. Okay, so it's e to the n times P times h. Okay, the second application is with respect to equilibrium states. So if you came to von's lecture last week you already know what equilibrium states are or to Pablo Carrasco's lecture which he also defined equilibrium states. So Bowen was able to use this symbolic model to construct to prove that each holder potential has a unique equilibrium state. In particular there is a unique measure of maximal entropy for this class of systems. Okay, and also you have very good ergodic properties of these measures is that they are either Bernoulli or they are Bernoulli times a finite permutation which I call here a rotation. Okay, so this is Bowen 1970, two papers 1970 and 1975. Okay, so what happens in the case of flows? Again let me assume that I have an axiomade transitive positive entropy flow and I assume it has no fixed points. Let's assume, well you only need it not to have fixed points in the no wandering set. Well the conclusion is the same as Bowen got here but it's a little bit more complicated to prove is that phi has unique equilibrium states for each holder potential. Again this implies that it has a unique measure of maximal entropy and you can further get ergodic very strong ergodic properties of these measures which is to say that they either define a Bernoulli flow or they are a Bernoulli flow times a rotational flow. So it's a flow on the on the unit circle. Okay, you can also get but it's much more complicated to get the counting of periodic of closed geodesic of closed orbits. So Perry and Policor they were able to to to prove that the number of closed geodesics of period up to t, so this this is what this number defines, all closed orbits of period up to t. It again grows sort of exponentially fast but you have to divide by t because when you have a closed orbit a closed yeah a closed curve you don't know what is the base point so you have actually a sort of a continuum number of closed curves. So as long as you divide by t well you get exactly the right rate which is given by the topological entropy. Okay, so these are the main applications. So now what I want to do is that I want to go a little bit further and no longer assume that my system has uniform hyperbolicity. What I want to assume now is that it's just asymptotically hyperbolic and what do I mean by that? Well let's see. So for the purpose of this talk I will always assume that I have either a diffeomorphism defined in a surface or a flow defined in a three-dimensional manifold and that's why the title has low dimension so because we only consider these two situations and if I have these two situations I can define the notion of a high hyperbolic measure. What is a high hyperbolic measure? Well first of all it is an invariant measure under the system that I'm considering and I will assume exactly this asymptotic hyperbolicity. I will assume that almost every point has two Lyapunov exponents different from zero. One of them I have to assume that is smaller than minus chi to take care of the unstable direction. The other Lyapunov exponent is bigger than chi. So then you ask me okay this is okay for the case of diffeomorphisms because I'm in dimension two I can have at most two Lyapunov exponents. What happens with the case of flows? Well remember that the direction of the flow no dynamics happens so the Lyapunov exponent in the direction of the flow is always zero. So this notion also makes sense for flows and what I'm saying the case of flow is that I have again three Lyapunov exponents one is smaller than minus chi the other is bigger than chi and the third one is zero is the full direction okay. This is exactly in the context of non-uniform hyperbolicity because almost every point has Lyapunov exponents different from zero whenever they can be different from zero okay. Okay good. So are there examples of such measures? Well yes there are plenty of examples especially if you consider systems with positive topological entropy. Why? Because if you know that your if you if you have an initial measure which you a priori don't know that it is hyperbolic but you know that it is ergodic and it has entropy at least chi then the who else inequality which relates the entropy with the sum of positively Lyapunov exponents tells you that your measure is actually chi hyperbolic okay. So whenever you have positive topological entropy you know by the variational principle that you have many measures ergodic measures with positive metric Komogorov-Sinai entropy. If this Komogorov-Sinai entropy is bigger than the threshold chi here then automatically assuming it is ergodic you get that this measure is chi hyperbolic okay. So there are plenty of chi hyperbolic measures in the systems with positive topological entropy. All right. So what are the examples? Well the examples for the femmophism the most known example is that you get that cat map that we started with and you slow it down enough in the fixed point. If you want to know more about this you can watch Yasha Pessin's talk on Thursday right that he's going to to say to talk about exactly about equilibrium measures for this slowdown of the cat map which is nowadays known nowadays known as the cat talk map. Okay and in the case of flows this context of non-informal hyperbolicity includes geodesic flows on surface which I no longer assume to have negative curvature everywhere but just non-positive curvature okay. So if I have non-positive curvature and I want to rule out the case of the torus so if I assume that the curvature is non-identically zero when you consider the geodesic flow what you get is a flow with positive topological entropy. All right. An example is to get a surface with hygenos and just put a cylinder on it. There's a back measure on or the geodesic flow on this surface will have positive topological entropy. All right. Okay this follows this follows from a formula that that passing that passing has for the topological for the metric entropy of the back measure actually on this on these surfaces. Okay. So this is the context that I want to consider and as I as I told you here well you start to have uniform hyperbolic with the uniform hyperbolic case and how do you get uniform hyperbolicity as long as you have negative curvature for example in a manifold the geodesic flow is going to be uniform hyperbolic. If I relax the condition to have a non-positive and non-identically zero curvature I get a non-uniform hyperbolic but you could ask does it do there exist surfaces which have positive curvature in some parts but still you have positive topological entropy and the answer is yes. There are examples constructed by these guys here and how are the examples? Well the first one is in genome zero so you get this sphere and you blow it up into six points just like here and in each of these bow ups you put a cup you put this cup here and you consider the geodesic flow on the surface well this geodesic flow is going to have positive topological entropy. The idea is that a point a point walking through a geodesic here spends much more time in the region of negative curvature than in the region of positive curvature. So on the average asymptotically what you get is what you would get in the uniform hyperbolic case you get expansion and you get contraction okay you can also do that in the case of genius one for example you get the torus here and you blow it up in this number of points and in each of these you put a focusing cup you get this guy here well the geodesic flow on the surface is again going to have positive topological entropy all right so this class of non-uniformly hyperbolic flows is much larger than the class of uniform hyperbolic ones okay so what do I want to say now well I want to say what are the results that we are going to discuss in this mini course so the first one is again remember that I always assume that I'm in low dimension I'm either with a diffeomorphism on a surface or with a flow on a three-dimensional manifold so let us start with surface diffeomorphism what is the result the first result in the in this context of non-uniform hyperbolic surface diffeomorphisms well it is given by katok which used passing theory in order to get a surface diffeomorphism with positive topological entropy and construct harsh shoes with large positive large topological entropy okay unfortunately his methods did not allow in general to construct harsh shoes with full topological entropy this this difficulty was bypassed recently you actually it the paper was published three years ago by sarik which was he was able that given a c1 plus beta surface diffeomorphism so you assume this regularity you assume that the derivative is better holder so given a c1 plus beta surface diffeomorphism and given a threshold he I can construct symbolic a symbolic model for f what is that well remember that a symbolic model for diffeomorphism is a triple the first two coordinates define a topological mark of shift the second that the last one defines the coding which here we get that it is holder continuous and what are the main properties of this coding well first of all it has to be relevant for the dynamics so it intertwines the dynamics of my diffeomorphism with the dynamics of the left shift the second property is that it has to be relevant somehow it has to be coding a large portion of the dynamics and sarik was actually able not only to code one measure at a time but he was able to code with the same symbolic model all measures that are he hyperbolic so this is usually an accountable number of measures but he was able yes given a he I can code at the same time all measures that are he hyperbolic okay you should regard for now for the next 30 seconds that this is just the image of of sigma itself forget about this sharp here okay so okay we have a good coding but in order to be good or in order to to to get interesting dynamical properties of the map below I should be able to get this symbolic model above in a way not to increase a lot the complexity of the system that I start below because I could increase a lot the complexity and well knowing things about the complexity above would not tell me anything about the complexity below so in order to get this no increase in complexity it is very important to know finiteness to one properties of this extension map pi so in the case of sarik's theorem he is able to get that for all points in this image here when you look at the number of pre-image that you have above this number is basically finite so the number of pre-image again in this sharp set here is finite so morally you are you are constructing a symbolic model that is finite to one and you know by Rohlin's formula that finite to one extensions do not increase the entropy so what you did is to build a symbolic model above that does not increase the entropy of the system that you started with in particular whenever you start in particular you can do many things one of the things that you can start with a measure here and lift it above in a way that the entropy of the measure above and the mat and the measure below are exactly the same so this lifting property which is usually not satisfied by extensions guarantees to you that measures of maximum entropy below are somehow related to measures of maximum entropy above so you can get some information about measures of maximum entropy for example so this is the result of Sariq and just to complete what is the sigma sharp the sigma sharp is what is called the recurrent set of sigma generally the sigma is a topological Markov sheath so it's given by an oriented graph unfortunately what happens in the non-uniform hyperbolic case is that this oriented graph is usually going to have countably many vertices it will not have finitely many vertices so it makes sense to define what is this recurrent set and what is the recurrent set well is the subset of paths on your graph that have a vertex that repeats itself infinitely often in the future and has a vertex that repeats itself infinitely often in the past well this is a subset of sigma but it's a good subset of sigma because you know by Poincare recurrence that all measures that are supported on sigma shift invariant they give full weight to this sigma sharp right so for the purpose of measures this is a cheap condition to impose and imposing this condition allows us to prove exactly this finiteness to one property okay so this is the result for surface difelomophisms that is another result which is still in the case of not of difelomophism but maps which is symbolic dynamics for non-uniform hyperbolic billiards so what is a billiard a billiard is let us assume that you consider a compact domain with piecewise smooth boundary call it t and inside this t you can consider the billiard map and what is the billiard map let me draw for example t like this the billiard map is you can you start here at the boundary of t in some direction call this theta and let me parametrize the boundary by this parameter r and what you do is that you walk at unit speed inside this table which you call the billiard table until you hit the boundary again when you hit the boundary of t again what happens is that you you have a specular reflection and you continue walking so what happens with the these angles here is that theta prime here the angle of incidence is equal to the angle of reflection okay this is the billiard map and where does it live well well I didn't say it is a map f that lives on the boundary of t cross they allowed angles minus pi over two pi over two and it sends a point here to another point which is exactly the next hit on the billiard table okay so these maps they have a natural liuville invariant measure which is given by this formula here okay r is the is the parameter called in the the boundary theta is the parameter called in the angle so mu given by this formula f invariant and well you are in the case of a surface because the boundary of t cross an interval is a surface is actually a cylinder so you could wonder why not applying the result of sariq directly to this to the setting if you have for example that this measure is has positive entropy well unfortunately that map f billiard maps they are usually not continuous why because you have these breakpoints here and you also have what are called the glancing orbits that an orbit could be tangent to this region here so tangencies and singularities of the curvature guarantee to you that this map is not a diffe amorphism okay and also its derivative is not bounded so you are in a much more complicated situation than in the case of surface diffe amorphisms but nevertheless it is possible to adapt the methods of sariq in order to get a symbolic coding for these these systems as well so the assumption is that if this measure mu here is ergodic and if it has positive entropy well ergodic plus positive entropy what did we learn by well's inequality we learned that it is hyperbolic so you have this measure mu being hyperbolic and what you can get is that you can get a symbolic model for these billiard maps so you get a pair sigma sigma and a coding such that the same properties as I mentioned before hold it intertwines the dynamics it is relevant for the my reference measure and it doesn't increase the complexity of the system that I started with okay so what are the examples that you have these two properties here ergodicity and positive entropy well you have for example the sinai billiards which are also called dispersing billiards in which the boundary is made of convex curves but you also have what are known as billiard uh bunimovich billiards which are the union of segments and pieces of circles which define a non-uniform hyperbolic billiard for example this one which is the billiard table with pockets this one which is probably the most known one due to the physicist which is called the be bunimovich stadium and this one which is called the flower okay in all of these billiards if you consider that environmentally we'll measure this measure is going to be ergodic this measure is going to have positive entropy so you can apply the theorem for that case for for these cases all of these cases okay so to finish today's introduction well actually almost to finish today's introduction I just want to mention what is the result for that we will discuss later for non-uniform hyperbolic flows in three dimensions so we start with this three-dimensional manifold consider a flow actually first consider a vector field so let's let us assume different from the the case that we were assuming previously that our vector field is different from zero everywhere before in the uniform hyperbolic case I was only assuming that there are no fixed points in the no wandering set now I really want to assume that there are no fixed points anywhere okay so given this vector field that defines a flow I call it phi okay and I want to consider a hyperbolic measure so consider this hyperbolic measure mu what is the result in in collaboration with sarig well is that this triple m phi mu has a symbolic model and what is the symbolic model for the flow remember it is a topological marker flow and a coding and well I want the good properties for the symbolic model as well what are they when you look at the suspension of unit vertical flow on sigma r and when you look at the flow on the manifold what the coding does is to intertwine these two flows okay good this coding is and I forgot an r here this coding is relevant for the measure that I started with so the its image has full measure and also it doesn't increase the complexity of my system so for every point in this image here the number of pre-image that it has is finite okay and again what is this sharp thing here well it is the recurrent set of this suspension space which is given by all pairs vt where the symbolic symbolic component is in the recurrent set of my symbolic space so it has a vertex that repeats it infinitely often in the future and a vertex that repeats it infinitely often in the past okay good so observe the difference from this result to Sarig's result here we can code one measure at a time Sarig was able to code all he hyperbolic measures simultaneously okay nevertheless we can still get all the the the let's say almost all the consequences that symbolic dynamics gives to us so to finish today I will mention the main applications of these theorems of these last three theorems that I told you and let me start with the case of diffeomorphisms the first one is counting of periodic orbits you get exactly the same sort of exponential growth of course since you are in a very general context which is that of non-uniform hyperbolic system nothing prevents you that this number is actually infinity you could have a piece of the dynamics which you have the identity right so the only thing that you can get is actually a lower bound on the number of periodic orbits so this is what's given by Sarig in the same paper that he constructed the symbolic models and in the same paper he was also able to prove that these these surface diffeomorphisms they have at most countably many measures of maximal entropy okay basically because the symbolic space that you construct has that property so you can project the symbolic properties below recently this has been improved by Bozik, Rovisie and Sarig which is that they prove that if you assume a further regularity of your system if you assume it to be infinity and a very cheap topological assumption which is dynamical topological assumption which is of being transitive then actually you can only have one measure of maximal entropy you always have at least one by a result of new house but under this infinity transitive assumption you do not get more than one okay their proof is related to to this result of Rodríguez, Rodríguez, Tazibi and Ures which they prove that surface diffeomorphisms they have at most one SRB measure okay so and also Sarig got observed that the publication year was before the year that he constructed the symbolic models but this is a flow of the system of referees but anyway he was able to use the the the symbolic model that he constructed here in order to get ergodic properties of the equilibrium states of equilibrium measures so if you assume that you have an equilibrium state of hold the potential with positive entropy then your system is either a Bernoulli automorphism or a Bernoulli automorphism times a finite rotation is exactly the same result that I mentioned of Bowen in the case of uniform hyperbolic systems okay so there is also this result of Boyle and Bouzou which discusses the almost Borel structure of surface diffeomorphisms I will not say anything other than that if you want you can look at the archive okay so but now let's go to the case of flows and in the case of flows if you have a three-dimensional flow with positive topological entropy what consequences can you get from the existence of symbolic models well again we can count periodic orbits and the counting of periodic orbits is again only one side that you can only get a lower bound and it again is given by the same sort of formula it is exponential in the entropy and you have to divide by T and here I forgot to add that this result only holds if you assume that your flow has a measure of maximum entropy so in parentheses here I should have put assuming phi has a measure of maximum entropy which happens for example in the case that you are c infinity all right good so the other result is that you again can get at most countably many ergodic measures of maximum entropy again because the symbolic model has this property and the final result is a result in collaboration with Francois Le Drapier and Omri Sarig which is the count the the same version of this result of Sarig 2011 for the case of flows so if you have an equilibrium measure of a whole depotential with positive metric entropy then the system it generates is either a Bernoulli flow or a Bernoulli flow times a rotational flow okay so for today I think what I want you to to remember for tomorrow is that what is a symbolic model and why are they important for the purpose of getting dynamical consequences okay tomorrow we'll start discussing Bowens construction Bowens classical construction of pseudo orbits and then we'll go further to Sarig's construction for non-uniform hyperbolic surface defamophism all right so thank you thank you for your time