 As you might have seen, you received an email saying that we should be on time today because we have to leave the room at 11.50. So I want no presentation. And so the goal of today is to discuss a little bit symbolic dynamics for non-uniformly hyperbolic surface defilmophisms. And it is to discuss the result that I mentioned on Monday, which is Sarek's theorem that says that if you get a C1 plus beta surface defilmophism, given any chi positive, you can find a universal topological Markov shift and a universal coding which intertwines the dynamics of the defilmophism with the dynamics of the shift map that is relevant for all measures that are chi-hyperbolic simultaneously, and that has the finiteness to one property. Okay? So forget about these sharp things and so on. The goal is to, at least in the level of non-uniform hyperbolic dynamics, to discuss a little bit how to construct a symbolic dynamics. Okay. So before that, I would like to remind us of what we did yesterday. So yesterday, it's right here. Good. Yesterday we discussed the method of pseudo orbits due to Bowen, and we can divide it into four steps. The first step is finding good coordinates for the system, and these coordinates were given by the Lyapunov charts, which in particular have the property that if the orbit of a point, if the image of a point is close to another point, then you can consider the map f with respect to these coordinates at x and at y, and this change of coordinates, it gives you a map fxy, which is hyperbolic-like. It is close to hyperbolic matrix in a uniform domain of size epsilon. That allowed us, after, first of all, the size of the domain is uniformly bounded from below, so it is enough to get a finite set of these charts that are delta dense in the manifold, and that allows us to apply graph transform methods in order to construct an infinite to one extension, and how is the infinite to one extension given? Well, you constructed this finite set of Lyapunov charts, so the set of vertices is going to be exactly the set of these Lyapunov charts, and you are going to say that you have a transition from one Lyapunov chart to the other, whenever the image of the center of one chart is close to the center of the other, and vice versa. And when you have that, well, this map fxy and its inverse are going to be hyperbolic-like, and being hyperbolic-like you can apply the graph transform method, and so you can construct this map pi that to each sequence sigma of Lyapunov charts like that, which is nothing but absolute orbit because the requirement here, the nearest neighbor condition is just that the image of one point is close to the next point, so an element here in this symbolic space is just a pseudo-orbit, and the graph transform method allows us to each pseudo-orbit to associate a point which is given by the intersection of this S-curve with the U-curve. This is the unique point that is shadowing the orbit of these Lyapunov charts. Good. So we saw also that this map is usually infinite to one, so we need to do a further step, which is the step of Bowen's and I refinement, and how do we do that? Well, this map pi induces a good cover on the manifold, and this cover is given by the projection of the zero cylinders in the symbolic space below, so each zv here is the projection of the zero cylinder with position v. So this is going to be a cover in your manifold, and whenever you see a non-trivial intersection, how do you destroy them? You destroy them dividing one of the elements of this cover dynamically into four pieces. Then you consider the refinement of all of these partitions. This refinement is going to be a mark of partition that is going to define to you a new finite one symbolic model. So this is the summary of yesterday, and what we want to do today is to try to do more or less the same scheme for the case of non-uniform hyperbolic dynamics. So the first difficulty is step number one. How do we generate good systems of coordinates on which your difomophism looks like a hyperbolic matrix? Well, as I told you yesterday, the case of uniform hyperbolicity is very good because everything varies continuously. But the sole definition of non-uniform hyperbolicity is an almost everywhere statement. So in general, you do not have things very continuously. At most they vary measurably. So we cannot do exactly the same thing as we did before, but we can do similar things. So the step number one, before the setting that we are going to work today is just to leave it here in the blackboard, is that of surface difomophism, which you assume a regularity C1 plus beta, and you want to code he hyperbolic measure. So you are given a positive he, and you want to code he hyperbolic measures. So what is the definition of non-uniform hyperbolicity? As I told you, it's an almost everywhere statement. And how am I able to code all he hyperbolic measures at the same time? Well, I'm not going to look at measures, but I'm going to look at points that have a good enough hyperbolicity. So you get a point in M for which there exists vectors, unit vectors in the tangent space of M. And the requirement on this point is that the limit, so the Lyapunov exponent associated to this S direction is smaller than minus he, log of Dfn Esx is smaller than minus he. You can actually take this limit for both future and past. And the same thing for the EU. But now you assume it to be at least he. If we are able to code all of these points with this property, then we are able to code every he hyperbolic measure because every he hyperbolic measure is carried out by the set of points satisfying these conditions. In general, you do not code all of these points. You have to add some sort of recurrence condition, but some recurrence conditions are very cheap because you know that infinite measures, they are recurrent, so you are in good shape. But what I want to discuss is how to, from a point satisfying these two properties, how to get the equivalent of Lyapunov charts in the non-uniform hyperbolic case. Well, these charts are known as passing charts. And in order to define the passing chart, first I would like to introduce three parameters, which are the parameters s of x, u of x, and alpha of x. And what these parameters tell us is how good the hyperbolicity at the point x is. So s of x is this weird sum here, and u of x is something similar, and alpha of x is just the angle between s and u. So first of all, note that this is a sum that is finite. Why? Because you know that the limit, the first limit here is smaller than minus 3. So this goes to 0 exponentially fast, so you are in good shape, this is finite, and this square root of 2 here is just made so that this number also is bigger than 1. So this belongs to 1 infinity, and the same thing for this number here. What are these numbers measuring? Note that whenever s of x is big, it means that it takes a long time for you to see the contraction in this direction. So in some sense, s of x is measuring how good the hyperbolicity in the s direction is. The worse the hyperbolicity in the s direction, the bigger s of x is. The same thing for u of x. u of x measures somehow how good the hyperbolicity in the u direction you have. So this number is big whenever it takes a long time for this to go to 0, so I'm saying exactly that it takes a long time for me to see the contraction for the past in the unstable direction. And alpha also measures somehow how bad or good the hyperbolicity is, because whenever s and u are very close, this means that it's very hard for you to distinguish between the stable and unstable directions. So in the uniform hyperbolic case, you do not have that. You have a uniform angle bounded away from 0 and from pi over 2, for example. So these three parameters, they somehow measure the hyperbolicity of your point. Okay, why? Just to make this to be bigger than 1. Okay, so as you remember Lyapunov charts, we first defined a linear change of coordinates, right? A map from R2 to the tangent space. So I'm going to do the same thing here. So I'm going to let c of x from R2 to txm be the linear map that is sending the first canonical vector. Okay, remember what we did yesterday? We sent the first canonical vector to the stable direction and we sent the second canonical vector to the unstable direction. And this was good enough for us to see that the derivative of the map in these coordinates is hyperbolic-like. In this case, this is not going to work. Why? Because it should take into account how good or bad the hyperbolicity in this direction and in this direction are. So what you do is exactly to introduce these two parameters s and u here. You divide by s in the first vector and you divide by u in the second vector. And these guys are saying what? They're saying that whenever you see a bad hyperbolicity, c of x is taking a vector which is big to a very tiny vector. Or reversely, it's getting a very tiny vector and it's zooming in to a very big vector. They are zooming in the hyperbolicity in order to see uniform hyperbolicity. And this is actually the case. So you have this lemma here which is called azeladets. Azeladets, passing reduction. That tells exactly that when you look at the derivative of the map in these coordinates, you see a hyperbolic matrix. So what is it to say that you see a hyperbolic matrix? Is it smaller than e to the minus g? So you see the g coming up here. This is smaller than one. And you see b bigger than e to the g. So this map with these changes here are good enough for us to see the derivative as a hyperbolic matrix. But lambda, which lambda? Huh? This is lambda, if you want. Oh, this is lambda. Smaller than one. So we did the linear part. We were able to see the derivative of the difomophism as a hyperbolic matrix. Now we want to see f itself as a hyperbolic matrix, right? So define the passing chart at x. As a map, define a domain that I'm going to tell you soon. But the way that we want to define is just like yesterday. We want to get this map c of x and compose with the exponential map. And our first goal is to see that that map f of x that I defined yesterday which is the map f in this system of coordinates is close to a hyperbolic matrix. Well, as you see, what happens here? Psi f of x minus one has the inverse of this exponential of f of x minus one c of f of x minus one. What happens if you have a bad hyperbolicity? When you have a bad hyperbolicity is exactly what I said here. You are sending a very big vector to a very small one. Or reversely, the inverse of the map is sending a very small vector to a very big vector. So whenever you see bad hyperbolicity the norm of the inverse of this map is going to be very big, right? So when you compose it here you are going to see a big distortion. When you see a big distortion it is hard for you to compare f of x with its derivative at zero because you can change a lot. You know that f of x at zero is equal to this guy. In order to get that f of x in a small neighborhood is close to hyperbolic you should know how much f of x is getting distorted. It is getting distorted because you have this inverse here as much as the inverse norm of this guy is big or small. So this is the main problem. The main problem is that contrary to the uniform hyperbolic case the size of the passing chart depends on the hyperbolicity of the point. The worse the hyperbolicity the bigger is this inverse norm. The smaller should be the size because you see a bigger distortion. And actually you can see that the inverse norm of c of x you have an explicit formula for c of x so you can calculate the inverse norm and actually it is not the inverse norm it is the Frobenius norm of this inverse is directly associated to the hyperbolicity parameters s, u and alpha by this formula. So this means exactly when you have bad hyperbolicity you see that either this number or this number is big or this number is small so this is reflected by saying that this number is big and when you see this number big you should consider the passing chart in a smaller domain and that is why we introduced a parameter Q of epsilon and it is going to be exactly what remains here in the definition of passing charts so it depends on this norm and it depends how well I put a very negative exponent here so you see that whenever this is big this number is small because it is a big number to a very negative power and I can actually put an epsilon to the 3 over beta here and this epsilon to the 3 over beta has two purposes first to get a uniform bound for the capital Q and second it is for when you do the calculation sometimes you want to absorb constants that appear in the calculation and this number here is going to allow you to do that so this is... okay so we defined the size of the passing chart and now these guys are good enough what? 21 plus beta so now with this definition f of x is close to the same hyperbolic matrix that we have as a lattice passing reduction so we are in good shape yes okay so this is lemma 2 from yesterday and now we want to prove lemma 3 what is lemma 3? well what we want to do now is not to put here f of x but to put a point nearby to f of x right? and this is the first difficulty that we encounter and Sarigi introduced the notion of epsilon overlap and what is that? well it is exactly a requirement to guarantee that this map is close to a hyperbolic matrix so what we want now is to find conditions on x and y for which when I change f of x to y here I still get the same result as I wrote there so what conditions guarantee f of x, y to be close to a hyperbolic matrix? before we only required that the point f of x is very close to y why? because these things they are very continuously so whenever they are very continuously this thing can be seen as a perturbation as a perturbation like this of the map f of x itself and f y is close to f of x then this is very close to the identity and we are in good shape but now what happens with this composition here? in this composition what is going to appear is the composition of c of f of x with the inverse norm of c of y and in principle because things only very measurably having y close to f of x does not guarantee that c of y is close to c of f of x and that is a problem so the way that we define if y is close to f of x not necessarily c of y is close to c of f of x and then I would have a perturbation that is very big this is not close to the identity so how did Sari deal with this problem? he said ok don't worry I am going to restrict the possible edges that I have and I am going to say that I can pass so right that I can pass from one charge to the other if I need to have the proximity of the points so this is very small and the same thing for the inverse but I require more I require also that c of f of x is very close to c of y and I am not going to put the quantifiers here and c of f of minus 1 of y is very close to c of x and when you have these conditions then a transition from one charge to the other guarantees by definition that this map is a small perturbation of a hyperbolic map so it is again hyperbolic like ok? so because the work of Sari is long I decided only to focus on two main steps of the proof and to me this is the first one this is the first good definition that he introduced in order to guarantee that you have this hyperbolicity here ok so we are able to define what passing charts are and as we saw here with this definition of epsilon overlap we are able to guarantee that this change of coordinates here is hyperbolic like ok? so what we try to do now is to continue with the sketch of proof of Bowen in this non-uniform hyperbolic case what are the difficulties? ok so coarse graining is before we wanted to press to a finite set of Lyapunov charts but why did we have finite because the sizes of the Lyapunov charts were uniform so we only require finitely many to cover the whole manifold in this case as you see you have bad points points of bad hyperbolicity in which this size can be very small so it is usually not possible to get finitely many passing charts to cover the whole manifold but you can still get countable how? you just get a countable and then set in the space of all passing charts so you pass to a countable and then set to the passing charts and these are going to be enough for you to code all points x that we are considering here ok? so let's keep going with the sketch of proof the next part is to construct an infinite to an extension using the graph transform so what is the topological mark of shift? again the set of vertices is going to be the set of passing charts that we constructed this is a countable set now but no problem and the set of edges is going to be the set of transitions from passing charts with these conditions here with these properties which are called epsilon overlap so once we have this topological mark of shift well just observe again that in particular an element here is again a pseudo-orbit because you are assuming these two properties here but it's more than a pseudo-orbit because you are assuming further two properties but never mind so the idea is to do what? is to apply the graph transform method in order to for each pseudo-orbit to associate the point right? so remember from yesterday how is the graph transform? the graph transform works in the uniformly hyperbolic case so the graph transform f of x y s and f of x y u what did they do? they they were associated to an edge so whenever you see a transition from Lyapunov charts yesterday the sizes of this Lyapunov charts were uniform of size epsilon so what could we conclude when we got a curve that is almost vertical here? well we could conclude that its image under this map f of x y would certainly go all the way from top to bottom and then we could restrict ourselves to the part of the curve that is inside this square and define this to be the unstable graph transform that is how we define the graph transforms well now we no longer know whether the size of the chart actually let me just put f of x here the graph transform that goes from x to f of x now in order to implement the same idea, the same method of graph transform we require that the domain of the passing chart at x and the domain of the passing chart f of x are comparable, are close to one otherwise you could be faced with the following problem that if, for example, q epsilon of x is much smaller than q epsilon of f of x then what would happen much bigger, sorry then what would happen is that this is the domain of the passing chart at x and this is the domain much smaller this is the domain of the passing chart at x and this is the domain of the passing chart at f of x so what happens with the image of a u curve here it would not go all the way from top to bottom so you need a good control on the ratio of the sizes of the passing charts in order to implement the graph transform and this can be done if the capital q of epsilon have a good property and what is the good property is that they cannot go to zero exponentially fast so lemma, the limit of 1 over n log of q epsilon f n of x as n goes to plus minus infinity is equal to zero almost everywhere and because of that I can define a better size for my passing charts and I will call this size small q epsilon and how do I do that? it is a formula that's relating the rate of convergence, the expansion and contraction that I have with the sizes of the charts and this sole property here guarantees to you that this number is positive it is by definition at most the capital q but now I have a good control on the value of it at f of x with the value of it at x what happens is that this belongs to e to the minus epsilon e to the epsilon if I take epsilon to be small this number is very close to 1 and this picture doesn't occur so the hyperbolicity that I have which is at least e to the r if I take epsilon very small will be bigger than the possible distortion that I see for these small q's conclusion is that this property guarantees to you that you can define a number potentially much smaller than the capital q on which you can apply the graph transform just apply the graph transform for curves of size ok? so ok we are good here I put that you also need some strong assumptions on the definition of s and u curves you can look at the paper if you want to understand that but never mind so what the graph transform gives to us is this map it is an if-need-to-one extension of our map f and what is the last step? the last step is the bow and sine refinement the cover that is induced by this map pi so you look at the projection of zero cylinders of the symbolic space to the manifold you get many sets like this zv so this curly z here is going to be a cover of your manifold not of your manifold but the relevant points that you want to code and what you want to do is to destroy non-trivial intersections so for every pair of rectangles in this cover you consider that dynamically defined partition of z into four pieces what are these partitions going to give to you? well you can consider the refinement of all of them and this is going to be a partition that has a mark of property so everything seems perfect end of proof well no that's when the nightmare comes why? because these mark of partition can be uncountable why? remember that the cover z is itself a countable partition a countable cover, right? it has countable many elements it might happen that when you refine a countable family you get an uncountable family for example consider the family of all intervals in zero one which have extremes in rational points this is a countable family of intervals but the refinement of all of them is given by points which is uncountable so we have to be much more precise to guarantee that this R is countable and what is the property that we require for that? the property that we require is local finiteness to guarantee that R is countable it is enough our mark of cover z is locally finite and what do I mean by that? I mean that every element of z intersects at most finitely many other elements of z ok so in order to do that unfortunately passing charts are not enough to guarantee this property so what was the solution encountered by Sariq was to define a new object that somehow separates the future from the past and before defining them ok so why do you want to separate the future from the past? well you know that in the non-uniform hyperbolic case so non-uniform hyperbolic systems by definition the future and the past behavior can be very different and it has not been seen by this small q here because this small q is considered a minimum in the whole orbit right? I would be able to separate the future and the past and analyze the future separately from the past why? because in general non-uniform hyperbolic systems they have a different behavior in the future and in the past so why not instead of considering this parameter here I consider the one sided versions of them so let me introduce two parameters stable q epsilon and this is taking the minimum only in for future and the unstable q epsilon which is taking the minimum for the past can you just finish this thing here? yes? that I can make a different definition the vertices of my topological Markov shift are not going to be passing charts but they are going to be two sided versions of passing charts and these are going to be good enough to guarantee to you that the cover z is locally finite I am changing the construction yes so the way I understand the introduction of this new object is exactly by separating the future from the past so you have these two numbers and observing that these two numbers they satisfy a greedy recursion what do I mean by that? I mean that the parameter qs epsilon at x satisfies this equality here it is the minimum between two values and how do you understand that? you add the value of it at f of x and you multiply by e to the epsilon and you take the minimum between this and the maximum it could be which is q epsilon of x so in some sense you can also understand qs epsilon as a measure of the local stable hyperboleicity of your point and how good is the local stable hyperboleicity of your point at x? well it is as good as you have it at f of x and what you gain when you iterate backwards e to the epsilon then the maximum that it can be so similarly you also have that qu at f of x is as good as it can be how? well you see what is its value at x then when you iterate it forward you should gain some hyperboleicity e to the epsilon but this number cannot be bigger than the highest hyperboleicity that you are considering so these two properties hold and these two properties induce to us to consider the two sided versions of passing charts which we call here epsilon double chart so what is an epsilon double chart? it is just an abstraction of the passing chart but now having two parameters which you could see just as a pair of maps the restriction of this map to this domain ps squared and of the same map to the domain pu squared but the idea is that this map is going to take care of the stable behavior of the graph transform for the stable direction and this map is going to take care of the graph transform of the unstable direction so now we define new objects which are these epsilon double charts and we want to make sense of when do we have a transition from one of them to another so how to do that? how do we do that? well first of all you want to have an epsilon overlap in order to be able to pass from one to the other but you also want to relate these parameters ps pu qs and qu and what is the relation? it is exactly given by these greedy recursions so you require that the ps here is equal to the minimum of e to the epsilon times the stable parameter here and the maximum it can be which is the q epsilon of x and qu to be the minimum of e to the epsilon pu and q epsilon of y so now I define this new object I define how to go from one of these new objects to another one how well to have the old property plus this greed recursion involving the hyperbolicity parameters what is the motivation for this? you can see as it comes from the definition of the stable and unstable small qs alright? so with this new definition let us see what happens and now we have 5 steps in the method of the proof the first step is before it was the introduction of passing charts now it is the introduction of epsilon double charts which are these guys here you can just see as a pair of passing charts in different domains and well, you want to pass to accountable set of them no problem, you just look at the space of all epsilon double charts you pass to accountable and then set subset of them, good, step 2 is ok step 3 is to get an infinite one extension before we were already able to do that so we do exactly the same thing but now our topological Markov shift the vertices of it are going to be epsilon double charts and the edges are going to be that definition over there so you should have epsilon overlap of the two passing charts and maximality of these four hyperbolicity parameters and the maximality is given by those two equalities over there ok? so what does this allows you to do? allows you to consider the topological Markov shift and now I call a sequence on my graph a path on my graph to be an epsilon GPO, epsilon generalize pseudo orbit, just a name so now from every epsilon GPO again using the graph transform, you can define a point in the manifold and it is given by the same identity that we had before which is this identity here but what is the difference now? the difference is that the next step would be to consider the cover Z and then to refine it what we required and I wrote it over there have Z to be locally finite so now we have a step three and a half which is called an inverse theorem and what is the inverse theorem? it is exactly telling you when the map pi that we constructed here can lose the injectivity so what do I mean by that? I mean that whenever you have two epsilon GPO's that have the same image under pi, you wanted to be able to prove that the parameters of one is close to the parameters of the other such that whenever you fix one of these guys, you do not have many options for these guys and this is somehow saying to you that your marker of cover is locally finite well, the theorem that Zarek proves is the following you have to require a further assumption which is that these two sequences are recurrent in some sense you are just saying that the points are returning to a set of good hyperboleicity bounded away from zero and this is very cheap from the point of view of ergodic theory and you wish to prove that the parameters of V are very close to the parameters of W and this is actually what Zarek proves in other words he writes V which is an epsilon generalized pseudo orbit so it is a sequence of epsilon double charts here and he writes W in the same way and he compares the parameters of these charts with the parameters of these charts well, the Xn is close to Yn we already knew because it is a pseudo orbit but these things are new now the stable parameter of the nth chart of V is close to the stable parameter of the nth chart of W and the same thing for the unstable parameter with this inverse theorem you are able now to apply the bonus now refinement what is the cover what is the Markov cover it is almost the same thing as before but now I only consider the projection of sequences that are recurrent why? because it is only in this space here that I have the inverse theorem so when I project only restricted to these guys I can conclude that this cover here is locally finite being locally finite I can refine it and not get something that is uncountable I get something that is countable so I do exactly the same thing whenever I see a non-trivial intersection I partition the rectangle into four pieces and then I consider the refinement of all of them now with the help of the inverse theorem I can prove that this cover here is locally finite hence this cover R this partition R is countable because in principle you could have an orbit that has pseudo orbits that are in some sense nested in this way you would get uncountably many you would not be locally finite because a point here would belong to infinitely many elements Z sub V and why this can happen this the size of this chart here can be very small because either the future behavior is bad when you separate the future from the past you are able to separately control it and show and then what you show is that these charts here are not going to be good for coding because I would have to have a maximality of either the stable or the unstable parameters okay you separate in order to control yes exactly you separate you can get a better control of it you are able to show that it varies up to a bounded arrow and then you are good and when you don't separate you cannot see that so I I mean maybe it's possible but I don't know a way of getting the construction only considering the small Q absence here because it can be bad for many reasons okay so now we got a countable cover countable partition and the end of proof and the end of time thank you