 Okay, so today I want to discuss a little bit about one of the ways of constructing symbolic dynamics, symbolic models for uniformly hyperbolic defilmophysm and this is known as the method of pseudo orbits and it is the way it's presented here it's due to Bowen. The original method as I mentioned to you yesterday due to Sinai is called the method of successive approximations. Actually if you look at the website of the of the conference I put some lecture notes there that have much more than I will discuss here and in particular it has a small discussion about this method of successive approximations if you want to learn a little bit more about that. Okay, so what is the method of pseudo orbits? For simplicity let me assume that we have an anodized defilmophysm on a surface so let f from m2 to m2 be an anodized defilm and what do I mean by that? I mean that the tangent bundle of m has a that exists a continuous def invariant decomposition tm equals to es plus eu and there exists constants a c positive and a lambda between 0 and 1 such that when you get a vector in this es what you see is a contraction under the action of the derivative and what you see for vectors in eu is an expansion so such that dfn of vs is smaller than or equal to c lambda to the n vs and to see expansion for eu is the same thing as to see contraction if you pre-iterate your vector so df minus n of eu is less than or equal to c times lambda to the n vu and here is for every n greater than or equal to 0 and vectors vs in es and vu in eu okay so what I what this says is that at the linear level what you see is expansion and contraction what I want to to do now is to introduce charts that are more suitable for understanding the behavior of f first of all I want to see the same behavior the same expansion and contraction behavior for f itself not for only the derivative and also why do I consider charts because I prefer to work in a Euclidean spaces than in this abstract manifolds so the first step is it's actually a simplification for the proof is that I cannot assume that this c here is is smaller is smaller than one so I cannot assume for actually that this c here is equal to one so this is what we call an adapted metric and I will leave as an exercise to you to show that if you have an analysis of the film opheism then you can find an equivalent metric to the remaining metric that you are considering for which there's inequality for this new metric holds but with c equals to one so that exists an adapted metric and I can put like three bars here to to make the difference for the the metric that we are considering the manifold on M and what do I mean by that I mean that the same inequality holds but with C equals to one this is just to simplify the calculations that we are going to do soon the F or actually I can put I can call this star and I can say that star holds with C equals to one okay so for for now on we assume that the metric that we are considering is adapted it's adapted for the difilm opheism that we are considering and as I told you what I want is that I want to construct charts that are easier to understand the behavior of F itself and to construct charts the first thing that we want to do is to to make a change of coordinates for the derivative such that the derivative itself is a looks like a map from R2 R2 to R2 which is expanding and contracting so for that let me introduce a linear map so for every x in M call ES and EU be unit vectors in ES X and EU X so this is a unit vector in the contracting direction this is a unit vector in the expanding direction and we define a linear map from R2 to TXM by well to define a linear map we just have to say what it does with a basis of R2 I'm going to send the first canonical vector to ES and the second one to you so C of X getting this vector here and it's sending to well you have here the X M it's sending to ES X and it's sending the other vector to EU X okay so I claim that this this change of coordinates of the tangent space of M makes when you when you consider the map DF you what what you're gonna see is a hyperbolic matrix so lemma what it means to see DF X in the system of coordinates well first you iterate with respect to CX so you come from R2 to the tangent space of X then you apply the derivative so you come to the tangent space of F of X and then you have to apply the inverse of the linear transformation at the point F of X with C of F of X minus 1 so then you come back to R2 and the lemma is that when you write this thing what you see is a map from R2 to R2 is there is a linear map and it can be written like this where A is smaller than or equal to the constant contraction to the contraction constant lambda and B is bigger than or equal to lambda to the minus one okay so there are many things here first of all is that I'm saying that these two vectors are eigenvectors and the eigenvalues associated to them one of them is smaller than one in absolute value they all they're always bigger than one in absolute value and the proof is actually quite easy you just see what this composition makes what how this composition acts on the vector so what that happens when you apply first C of X here well by definition I'm sending it to ES of X and what does the derivative do to ES of X well I know that the decomposition is the F invariant so ES of X has to go to the direction of ES of F of X right so it is a multiple of this how much well you apply this you take the absolute value of it and what you what remains is this direction so the image of this is this and then when you compose with the inverse of CF of X what you get is that this vector here goes back to one zero so the multiple of this vector goes to the F X ES X one zero no conclusion one zero is an eigenvector and this is the eigenvalue by assumption since my metric is adopted this thing here is smaller than or equal to lambda this is equal to a the a over there and this is smaller than or equal to lambda and this is the proof okay so what we did is that at the linear level we are able to find changes of coordinates that make the F look like a hyperbolic matrix but what I want to do now is to is to get the same thing for F itself not for the F so for that I have to find change of coordinates that come from R2 not to the tangent space but to the manifold itself correct so how do I do that well since I'm already in the tangent space how do I go to the manifold I can just consider exponential maps to assume or take epsilon is small enough such that all the exponential maps are well defined in a small rectangle around zero in the tangent space of X so let this such that these guys are for example I can I can require that they are defilmophisms onto their images lip sheets constant at most two why because I know that the derivative at zero is the identity so if I take epsilon is small enough then I can control the lip sheets constant of these maps okay so I'm going to define what is known as the Lyapunov chart at X and it's just a map from R2 to the manifold and how do I do well I have a map from R2 already to the unit to the tangent space so I just composed of the exponential map so this map here is just the composition of C of X with the exponential map at X okay so my goal now is to pass from the linear behavior of the derivative to a nonlinear behave of the map itself we have the following so if epsilon is small enough or the following holds for every epsilon is small enough what it is to see F in this system of coordinates well it is first to compose psi X so you come from art from a subset of R2 to the manifold then you compose with F then you compose with the inverse of psi f of X so let me give a name to this map this map is f sub X so the claim is that if epsilon is small enough then this composition is well-defined in minus epsilon epsilon square and it actually it is a small perturbation of this composition here so what do I mean by that I mean that you can write this thing as a UBV plus some map H of UV where let's say the C2 norm of H is smaller than epsilon okay okay so why is that true well because when you calculate the derivative at zero of this guy what you get is exactly this so if you are taking epsilon is small enough what happens with the map itself the map itself is close to its derivative so you can control the error term okay so there is no secret here what actually turns out is that you can get the same result if you change f of X by a nearby point Y so this is important for what you are going to discuss soon which is the graph transform method so actually I can get a better result than this so let me put lemma this is lemma 1 lemma 2 and this is lemma 3 the following holds for delta is small enough so if now I want to to make a change of coordinates that does not go from X to f of X but goes to X from X to a nearby point to f of X so if the distance of f of X to Y is smaller than delta then when I see F in the coordinates X and Y and let me give a name to this let me call this f of X sub Y this map here then this map is well defined can be written as f of X Y U V the same thing equals to a U B V plus some h tilde of U V and now in order to control this the behavior of this guy here I have to relax a little bit the derivative the norm that I'm considering here well for simplicity let me just assume that I consider the C1 norm so where the C1 norm of this h tilde is smaller than epsilon okay and the idea of proof of this is just seeing f of X Y as a small perturbation of f of X which it actually is because you can write f of X Y as f of X composed to psi Y composed to psi f of X minus 1 and f f of X is very close to Y this is almost the inverse of this so this is very close to the identity and if you have a control on the C2 norm of this and this is very small you can have a control on the C1 norm of this composition okay good so these are the charts that make f itself look like a hyperbolic matrix and I want to take advantage of this to construct invariant manifolds so how do we construct invariant manifolds if you have taken classes in hyperbolic dynamics you might have heard about the graph transform method so it is exactly the graph transform method that I'm going to apply now so the graph transform method and what is the graph transform method well I just show to you that f of X Y is close to a hyperbolic linear map so what happens if you come let's say that we are here in R2 and f of X Y is a map from R2 from a subset of R2 to R2 and let us say here that the horizontal is the stable direction the vertical is the unstable direction so what happens if I get an unstable curve here so what do I mean by an unstable curve is let us say that an unstable curve is the graph of a function that is very parallel that is almost parallel to the vertical direction okay and a stable curve is the same thing for the horizontal direction so what happens if you get an unstable curve and you apply f of X Y well if I didn't have this error term here what would happen is that this thing here would stretch vertically and would be a little bit closer to the Y axis right well the error term doesn't change the qualitative behavior of this so what actually happens when you iterate this curve under f of X Y is that you go all the way from the top to the bottom like this right and the same thing happens if you get a S curve here so let me get an S oops get an S curve here what is it it is the graph of a function that is whose graph is almost parallel to the S direction formally you can require conditions on the function that you are considering this function has to have leaf sheets constant at most one that's what you require to define an S curve so if you pre pre iterate this by f of X Y what you get is a curve that goes all the way down all the way down from the left to right like this right and what is the graph transform what is the graph transform method is just I'm going to analyze the map that is sending horizontal curves here almost horizontal curves here to almost horizontal curves here and almost vertical curves here to almost vertical curves here so if let me introduce annotation as you see in order to consider f of X Y and for f of X Y to have this hyperbolic behavior have to assume that f of X is close to Y but to do the same thing for the inverse I would have to assume that X is close to f minus one of Y so I'll give a name for this so right Psi X so right an edge from Psi X to Psi Y if exactly these two conditions hold so the distance of f of X to Y is smaller than Delta and the distance of f minus one Y to X is smaller than Delta and when you have these two conditions then you can get this behavior for U curves and the other behavior for S curves all right so these two maps are called the graph transforms so the stable graph transform of whenever you see these two properties here you can consider the stable graph transform is the map let me give X f of X Y S that gets an S curve at Y and sends to an S curve at X and similarly how does it send it? It sends applying the map f of X Y similarly the unstable graph transform is the map f of X Y U that gets a U curve at X sends to a U curve at Y so what do I want with these maps? Well the main property of them is the following what happens if you consider now not only one U curve but two U curves and you iterate and you apply this the unstable graph transform well again this guy goes all the way from top to bottom but what happens with the horizontal proximity of them? You are seeing a contraction in the horizontal direction so you expect that these two curves are much closer than these two curves right so you expect to see some contracting property for these maps this actually holds so the main property of the graph transforms which comes from the hyperboleicity they are contractions well and in order to say that something is a contraction I should say what is the distance that I'm considering when you look at the space for example of S curves at X each S curve is the graph of a function right so you can consider the distance between two S curves just as the for example the C0 the C1 norm or actually the C0 norm of the functions defined in the graphs so for each for each curve you have a function functions you can consider distances so this is what defines the distances in these spaces in the spaces of S curves and U curves and with this distance these two guys are contractions okay all right so what I want to do right now remember yesterday that the idea of symbolic dynamics is instead of describing the orbit of a point I want to describe some sort of a where these orbits of point belongs to what I want to do now is that I want to consider a sequence of points that's fine these two properties and given the sequence of points I want to come up with a new point whose orbit her orbit is shadowing exactly this fixed sequence of points so what do I mean by that well this is X minus 1 this is X 0 this is X 1 and if I have a sequence of points such that the distance of the image of this to this is smaller than delta and the same for the inverse here and so on then I want to find a point X I want to find a point X whose orbit is always belonging to this small rectangle around X I okay okay so let me give a name for the sequence of points so up sale the orbit is a sequence X n such that well since I already introduced the notation I'm going to use it psi X n goes to psi X n plus 1 for every n in Z okay yes well they ask yeah no that I can I can tell you that the yes exactly yes and I get rid of it I just delete it it goes all the way down from one side to the other I just delete it and I see what remains inside this small rectangle and how do I iterate this again well I forgot about this I iterate it again it goes a little bit all the way to the top and to the bottom I delete it and I keep doing that okay so I just see at this at the scale epsilon again an S curve is just the graph of a function whose lip sheets constants at most one when you apply this map f of XY the image of this goes to something that is beyond this rectangle so I'm defining what a map is it takes an S curve and if I apply f of XY it goes all the way beyond beyond these boundaries I delete these boundaries and I say that the image of my S curve is this is what remains inside the box okay I get rid of what is outside okay so given absolute orbit we can consider the following limits observe that if I want my point to be following the itinerary of this pseudo orbit well if I look at the time X n time n f of n of X should be inside this small rectangle right so what happens if I look at X n minus one well it fn minus one of X should be inside this rectangle so I know that if I iterate f of X n minus one X n to the inverse the image of this remains here but in other words what I want to convince you is that if I consider an S curve here and I apply the graph transform method it should come to an S curve here containing this point I'm trying to recover what should be the property of this point here well it should be in S curves all the way in all the charts that I'm considering so because of this contracting property I can actually try to find what is the horizontal direction of the point that I'm aiming at what is the horizontal direction at the point that I am in it I'm going to define as the stable admissible manifold of my sequence X n side X n and how do I do I take advantage of the contracting property of this so I go to the future time n and I look at any S curve at position n and I start to pre iterate this with respect to the stable graph transform so I go from X n to X n minus one by applying this then I go all the way to X zero X one each of these guys a contraction so you expect that this limit exists and actually this limit exists and it does not depend on the choice of S curves that you are considering in the future so this limit exists and it exists for some and actually for any choice of S curves ES n at X n for n greater than or equal to zero so what I just defined actually only depends on the future and what it is it is an S curve in the position zero and it tells me what is the position on the horizontal position of the point that I'm looking at I'm trying to construct so by the same reason I could define also the view of the sequence and what do I do I go all the way to the past I get a U curve and then I start to iterate the graph transform the unstable graph transform forward so I come from X n to X n plus one U dot dot dot up to f X minus one X zero and I take the limit as n goes to minus infinity so this limit exists for some and actually for any choice of U curves okay so how do I get the point that is following exactly the itinerary of this pseudo orbit I have an S curve and I have a U curve so I just get the intersection is a point that follows that shadows the pseudo orbit X n okay so I think I have ten minutes I think it's enough so what am I going to do now now I want to pass to a finite set of Lyapunov charts such that when I look at the space of all pseudo orbits generated by this finite set of Lyapunov charts I am able to to by applying this method here to recover all points of the manifold so this is what I'm going to call here coarse-graining and how do you do that well you fix a delta prime much smaller than delta it usually depends on the lip sheets constant of your defilmophism and take a delta prime dense finite set of vertices of points in M okay so I have a delta prime dense subset of M and what I'm going to do is that I'm going to consider all Lyapunov charts at these points to define an oriented graph g equals to V e where the set of vertices is exactly all the Lyapunov charts centered at points of my dense subset and the edges are they allowed transitions from psi x to psi y what I mean whenever the image of this is delta close to this and vice versa I write an edge from this guy to this guy okay this one so what happens is that the the passing the not the Lyapunov chart itself is two lip sheets so if you know that on R2 you are close to xn when you apply it when you apply the Lyapunov chart what you will actually get is this image here f of n of x is close to xn it's an actual orbit x is a point defined like this yes but because of the invariance of these guys what happens with f of x it's going to be the intersection of stable and unstable as well okay so what I want to do now is that what happened what what remains to to be done well we already constructed an oriented graph and we can consider the topological Markov shape defined by it but in order to get a symbolic model we need a coding the coding is actually given by this equality here so let's sigma to be the topological Markov shift associated to G and I'm going to define the map pi from sigma to M by doing exactly that equality over there so if you have a sequence of charts like this or an actual absolute orbit how do you define the point that you want to get in M well you consider the VS intersected with the view okay and what are the main properties of this map pi well pi is surjective why because if you want to to do something like this if you want to to realize this as a point x what do you do so if x belongs to M you find an xn in x such that the distance of xn to x to f of n of x is smaller than delta prime you can do that because you know that x is delta prime dance then you observe that this xn defines absolute orbit and because you have this proximity here the intersection of two of these two guys has to be x because this uniquely defines so this intersection is the only point that is delta close or delta close to this xn for all times okay so the second property is that pi is an extension of the map f so pi composed to sigma is equal to f composed to pi so our hope is exactly to find that this pi is the symbolic model that you are aiming at but unfortunately pi is highly infinite to one in general so pi is usually infinite to one why try to imagine what would happen so what what it means to try to check finiteness to one you fix a point in the manifold and you try to look at all sets of points xn for which this intersection here is equal to x right if by any chance you had here two options for xn then you would have uncountably many sequences uncountably many pseudo orbits for which this intersection is x so if this happens holds for two points on x then the pre-image of x is uncountable for each position I have two options so this is not the map that we are looking at that we are aiming to construct it is highly infinite to one in general so what do we do what we do is that we consider a refinement and this is the last step of the construction so it is a bow and refinement and the idea is that even though pi is infinite to one it defines a cover of my manifold by good subsets so what we consider is that let z be a family of sets zv where v belongs to the vertex set and what is the element zv so above you have good mark of properties because you are in a symbolic space so what we are what I'm going to do is that I'm considered all zero cylinders above and I'm going to project them below so z of v is going to be all the images pi of v underline and remember that this is an element of sigma yeah which I'm going to write now such that v is a sigma and v0 is the vertex v that I'm fixing so for each vertex I have a subset like this this is a subset of m this gives me a cover of m right it has a symbolic mark of property because it inherits this mark of property from above but it is not a disjoint code it is not a disjoint yeah it is not a partition of m so how do I get rid of non trivial intersections so for all z z prime z we partition or we dynamically partition z as follows so I have subsets like this and I want to get rid of non trivial intersections so in order to have the mark of property what I'm going to do is that whenever I see these non trivial intersections I'm going to divide z as follows you look at the stable boundary of z prime and you extend it all the way and you do the same for the stable stable and unstable so whenever you see this you just cut z into four pieces like that and formally what you have to do is that one element one of the elements of this partitions going to be the set of points whose stable piece does not intersect z prime but unstable does this one is the set of points whose stable curve and unstable curve do not intersect z prime and so on so that's why I say that they are dynamically defined okay you look at the stable and unstable fibers of the point inside the z and you see which of them intersects z prime according to that you divide z into four pieces so you have many partitions for z as you run over z and z prime here you can consider the finest partition or the the course is partition that refines all of these guys here so call this partition easy z prime this it is finite why because the vertex set is finite right my subset x is finite yeah so let R to be the partition that refines all of easy z prime as easy prime run over z so the claim is that this is the partition that we are aiming to construct so this partition has the mark of property and I know that I haven't defined this but if you came yesterday for the exercise class I did define it and one of the exercises was that whenever you have a partition with the mark of property you can define a symbolic coding and it defines a new topological mark of shift which I'm going to call sigma hat sigma hat and a new coding pie hat from sigma hat to M and the triple sigma hat sigma hat pie hat is a symbolic model for F or in summary what I did today first step was to consider systems of coordinates that see F as a small perturbation of a hyperbolic linear map then I took advantage of that to consider graph transforms what are graph transforms useful for they are useful to allow you to construct a first candidate for a symbolic model I constructed a topological mark of shift and a coding but this map pie although it is surjective and it intertwines the dynamics of F and sigma it is usually not finite one and when it's not finite one I'm not happy so what do I do well nevertheless it induces not a partition below but a cover that has the mark of property and once you have a cover with the mark of property you can apply a refinement procedure in order to destroy non-trivial intersections and get actually a mark of partition partition with the mark of property this new partition is going to give you a new topological mark of shift and a new coding and this new triple here is going to be the symbolic model that you are looking for okay so I think today I'm going to stop for now tomorrow we are going to try to apply the same idea for the case of non-uniform hyperbolic surface difomophisms and we are going to be faced with the difficulties that in general contrary to here here everything varies continuously in the non-uniform hyperbolic case everything only varies measurably so we have many problems many difficults count is that you are going to encounter and you are going to have to adapt the definitions all the way okay thank you