 Before I forget I should say that everything I will say is joint work with Keith Burns, Todd Fisher and Daniel Thompson. So this event is meant to be about, I believe it was algebraic, geometrical and probabilistic aspects of dynamical systems. I will not have very much to say about anything algebraic, but I will hope to say something meaningful about some geometry and about some probabilistic aspects of things. So I'll start by first discussing equilibrium states, then I will say something about geodesic flows and finally I hope to leave myself enough time to say something meaningful about non-positive curvature, which is where all the new results are. But first let me start by just giving some general background for the benefit of someone who's not quite an expert in this exact topic. So the setting is that we wish to consider some dynamical system, of course, by which I mean a compact metric space X and a continuous map F from X to itself. So this is our system. And if you have a system which exhibits some kind of hyperbolic properties or colloquially some kind of chaotic properties, then you know that it's not possible to make very good predictions. And so you end up treating observations of your system as instances of stochastic process, as a sequence of random variables. But if you want to say the words random variable, you need a measure. So the question arises, what measure should I use? So in general, I'm going to write Mf for the space of Borel F invariant probability measures on the space X. And this space may be quite large. So for example, suppose I have the simplest example from which everything comes, let me consider a map from the circle to itself. So X is just the circle, which you would think of as the unit interval with endpoints identified. And F is something conjugate to the doubling map. And let me assume that it's uniformly expanding. So you have this picture. I don't assume it's linear or anything. And now, of course, it's very well known that the space of invariant measures for this is extremely large. So if I want to do anything with it, what measure should I use? Well, the way that we determine which measure we should use is via the process of thermodynamic formalism. And so I have a continuous real-valued function on X, which I refer to as a potential. Then we may consider the topological pressure. The pressure is just the supremum, taken overall invariant measures of the entropy plus the integral of this potential. So of course, if you do the zero potential, you are getting the measure, you are maximizing the entropy, and you get the topological entropy. But you may have other potential functions. And the most important definition for the talk is that a measure is an equilibrium state if it achieves this maximum. So a measure is an equilibrium state. If this quantity, which is sometimes called the free energy or the negative free energy of the measure, is equal to the topological pressure. And the main questions that we will be interested in are the following. Do equilibrium states exist? Are they unique? And what are their statistical properties? By statistical properties, I mean, is it a Bernoulli measure? Does the measure satisfy exponential decay of correlations? Does the polynomial decay of correlations? Does it satisfy a central limit theorem? In fact, I will not talk about question number three much. And question number one will turn out to be not so difficult in the settings where we consider. So the real issue is question number two, uniqueness. And if I come back to this basic example, let me assume that F is at least C1 plus alpha. Then the following result is completely classical. Proved in the 1970s, in fact, for a much broader class of systems than just this, but this gives the flavor. You have the following things. For example, number one, if phi is holder, you have to have holder continuity of your potential. If your potential is just some arbitrary continuous potential, in general, we can say almost nothing. So we assume some regularity of the potential, but then you get the following. First, there exists a unique equilibrium state, which I'll denote mu subscript phi. And moreover, this equilibrium state is fully supported. The port is the entire system, which is to say it gives positive weight to every open set. The example one. Yes, I'm giving just the very, the very simplest example. In fact, you can say rather more than just that open sets get positive weight. You can give a very quantitative estimate of this, which is called the Gibbs estimate and but I don't want to formulate it because in our settings, if they come in the non-uniform settings, I will consider later it becomes much more complicated. But the point is that you have this unique equilibrium state and it has this fully supported property. Second, there's a very special, there's a very particular potential, which is in some sense perhaps the most important, which we may call the geometric potential. So if I let phi be minus log of f prime, then this quantity that I get for any invariant measure, just the integral, well, if mu is ergodic, then this gives you the Lyapunov exponent, or rather negative the Lyapunov exponent of that measure. And of course, in this case, you know that there's any relationship between entropy and Lyapunov exponent. You know that the entropy is always less than or equal to the Lyapunov exponent for a one-dimensional system like this. And that means the pressure is less than or equal to zero. This is just the statement that h mu minus lambda of mu is less than or equal to zero for every ergodic measure. Now in fact, you have equality for the topological pressure here. That does not follow from this inequality. That requires a little bit more proof. But one can show that in fact the pressure of this particular potential is equal to zero. And that if you consider the equilibrium state, the unique equilibrium state for the geometric potential, which is guaranteed by part one, because I assume f is c1 plus alpha, this unique equilibrium state is in fact absolutely continuous with respect to Lebesgue. And vice versa, it's actually equivalent to Lebesgue, and you can get some estimates on the density. So that if you are looking for a physical measure for the system, that is, if you are looking for a measure which on the one hand is related well to Lebesgue, and on the other hand has the invariance property, then this is your measure, and you can find it using the thermodynamic formalism. You can find it as an equilibrium state. For more general, all of this as I said holds for more general classes of systems, in particular holds for SRB, for uniformly hyperbolic axiom A systems, and then this measure becomes the SRB measure. But I will not discuss this. The last things I want to say in this setting are just that the function, the real valued function, which takes a real number q, and gives you the pressure of q times the geometric potential, this is a real analytic function. And it looks like this. We proved that, or we didn't prove I claimed that at q equals one, this is equal to zero. At q equals to zero, this is just of course the topological entropy, which in this case is log two. And it is a strictly convex, in this case it's probably strictly convex, but in general it's just a convex real analytic function. So I want to say one more, and four is just the answer to these statistical properties, which is that in this case, it satisfies exponential decay of correlation, central limit theorem, and the Bernoulli property, but since I will not discuss these more, I'll just say that. So this is the result, which is well known in the uniformly hyperbolic case, but of course one wants to understand non-uniform hyperbolicity. So one wants to understand what happens when you don't have uniform expansion, and the simplest example of this, in fact write the example over here, is the so-called Mandel-Pomomap, which looks exactly like the first example, except that at the point I take it to have derivative one. So that means that it's not, the fixed point at zero is not uniformly repelling, and now I assume that the near zero, the derivative looks like one plus x to the alpha or something like this, for alpha between zero and one, and now I may ask, what does some version of this theorem hold? And let me immediately observe that it cannot hold in exactly this form, there's an obstruction. There's an obstruction to part two here. Let me consider the geometric potential, and let me consider the particular measure, delta naught, which is the delta measure sitting at the fixed point at the origin. This measure has zero entropy and it also has zero Lyapunov exponent, precisely because of the indifference of this fixed point. So this is a measure with entropy zero, Lyapunov exponent zero, and therefore for the geometric potential, its free energy is zero, and that makes it an equilibrium state for this geometric potential, because this fact that the pressure of the geometric potential must be less than or equal to zero, that's a universal fact for maps of the interval like this. So this is an equilibrium state, but it's not fully supported, it's a delta measure, so we know, but this is a holder-continuous potential, so the theorem cannot hold exactly as stated. There's an obstruction, let me write down what that, in a little bit more general sense what that obstruction is. The reason that this is an equilibrium state, because the value of the potential function at this single fixed point is equal to the pressure. That's the obstruction, and the theorem is that that's the only obstruction. So here is classical theorem number two, slightly less classical than the first one. This first one goes back to the 70s to people like CNI, Ruel, and Bowen, many others. This one goes back probably to the 90s, and in example two, if I take a holder-continuous function, then I have the following. If that inequality, if I don't have this obstruction present that is, if the pressure of the system is greater than the value of the potential at the fixed point, so that delta naught is not an equilibrium state, then part one holds. There exists a unique equilibrium state, and it is fully supported. That obstruction is complete, in some sense, for part two. Well, you, in fact, there is an equilibrium state, the geometric potential, which is absolutely continuous, and there's another one, which is the delta measure. So there are two ergodic equilibrium states for the geometric potential. There's one, which is absolutely continuous with respect to Lebesgue, and then there's another one, which is just the delta measure. Sometimes this is called a phase transition, because you have two equilibrium states coexisting for the same potential. One also uses the word phase transition to describe the way in which number three fails here. Suppose I again consider this function, which takes the pressure of scalar multiples of the geometric potential. Then it looks like this. Just as before, it's, so this is, let me label my graph, this is the pressure of q times the geometric potential. On the interval, on the open interval from minus infinity to one, this is analytic. And then it has a point of nondifferentiability. So there's some kind of phase transition. So in ring number four, let me just say that in this case, when you have, when the obstruction is not there, and you have a unique equilibrium state, then all these things hold. It's exponential decay of correlations, it's central limit theorem, it's Bernoulli. If you consider, say, this absolutely continuous invariant measure, then it's a different story. It's polynomial decay of correlations, it's still Bernoulli. Maybe it satisfies the central limit theorem, maybe it doesn't. It's not alpha, but that's a different talk that I'm not giving. So the point of these, stating these two theorems is that I want to highlight the change in behavior that one might expect when you go from a uniformly hyperbolic to a non-uniformly hyperbolic setting and you consider the thermodynamic formalism. That was part one. I talked about equilibrium states. Now I should do part two. I should talk about geodesic flows. So all of this, of course, as I said, holds for more general classes of systems. The class of systems that I want to spend the rest of the time discussing are geodesic flows. So now, let me consider M. This is regular M, this is different from script M, which came before. This is a compact, smooth, I guess I better say connected, Riemannian manifold. So you have an ocean of geodesic. And in particular, if I have a unit tangent factor at some point, so then, of course, this projects to some point on the manifold, and this guy could determines a unique geodesic. So there's a unique geodesic on the manifold with the property that, at zero, it starts at the footprint of V and V is tangent to it at zero. So the picture is just, here's V and here's gamma of V. And the geodesic flow is the flow that you get by going from a vector on the unit tangent bundle, determining the geodesic that it determines, and then moving along that for time t. So the geodesic flow is a flow on the unit tangent bundle of M. This is important. The system that I'm going to consider does not live on the manifold itself. It lives on the unit tangent bundle. So this is G sub t, the one parameter. So it's a flow. It's a one parameter family of maps. And it takes a unit tangent vector V and takes the geodesic that it determines, move to time t along that geodesic, and you take the tangent vector. So if this is gamma V of t, then the tangent vector at that point is the image of V under the geodesic flow. It's a dynamical system. It's a flow. Now I introduced all the thermodynamic formalism for maps instead of flows, but it works in both. You can define it also for flows, and instead of, and so I consider the space of invariant measures. Given an invariant measure, I may consider the topological, or the measure theoretic entropy, which is just the entropy of the time one map of this flow. And then we ask about equilibrium states. Now, the first setting to consider is when the manifold is negatively curved. It turns out that in this case, you have a complete theory. So assume that all sectional curvatures of the manifold are negative. And in this case, one can do the following. Let me first say that locally, you can get a sense for what should happen by picturing a saddle. At some point, I was able to draw a saddle, and I'm not able to now. But the point is that if you consider how geodesics flow on this, you sort of expect them to diverge from each other no matter which way you're going. Whereas that's not going to be true on a sphere or on a flat surface. And that local divergence of geodesics, which comes from the negative curvature, is what gives geodesics flow on manifolds of negative curvature its hyperbolic behavior. And to make that a bit more precise, let me draw the following picture. Let M tilde be the universal cover. So for example, if I take a surface of genus 2, rightly object that if I embed this in R3 in the way this picture suggests, it definitely doesn't have negative curvature. So I just drew the topology. I didn't draw the geometry. The universal cover of this, Poincare disc, and the way to see the negative curvature is to view this as the octagon with edges identified. So you have a hyperbolic, you know, the picture of the Poincare disc here where geodesics are circles, arcs of circles, which intersect the boundary orthogonally. And if you take eight such arcs that are chosen just so, so that the boundary, so that the angle between them is pi by 4, then you can identify the edges of this octagon in an appropriate way and you obtain that surface of genus 2 with constant negative curvature. Now in fact, I don't want to assume constant negative curvature for any of the theorems. The negative curvature is allowed to vary. But when it's constant, you have this very nice picture. And in fact, in this setting, one could treat this kind of algebraically to the third word of the conference, but I'm not going to. The point is that once you look at the universal cover here, every geodesic on the surface lifts to a geodesic on the universal cover, which goes from one point on the ideal boundary. So the ideal boundary is written dy m tilde. And this point on the ideal boundary, let me call it psi, I can, let me draw a new picture so this doesn't get cluttered. If I take my point psi, and now I consider all of the geodesics which terminate, which are asymptotic in the forward direction to psi, well, you get this, you get this diameter, you get a lot of guys like this. You get some picture of this nature. And every curve that I just drew is a geodesic. And now I consider the foliation of the disc, which is given by taking circles that intersect these geodesics orthogonally. And this is, of course, the family of horror cycles. These are meant to be circles, but I'm a mathematician, not an artist, so please pretend they are circles. Each of these is a horror cycle. And if I consider a particular vector v, so v now is a unit tangent vector, then it determines a horror cycle. This is the stable horror cycle of v. And that's a subset of the universal cover. Now, given that horror cycle, I can consider, I don't think this is standard notation, but I'm going to write it. This is the set of all normal vectors to the horror cycle, unit normal vectors. That is, it's the set of all unit normal vectors that point in the same direction as v. And the property that these all have is that if I follow the geodesics they determine, all of those geodesics are asymptotic to the same point psi. Also, this is a subset of the unit tangent bundle. I'm conflating a little the manifold itself with its universal cover, and I'm not expressly denoting when I go back and forth between them. I hope it's clear. But I just defined a collection of points in the unit tangent bundle with the property that their forward geodesics are all asymptotic to each other. In other words, their forward orbits under the geodesic flow are all asymptotic to each other. Well, that's a stable manifold. So let me call this Ws of v. It's a stable manifold. H of s, the stable horror cycles, or in higher dimensions, the stable horror spheres, these determine a foliation of the universal cover. And the normal bundles to them, the unit normal bundles, determine a foliation of T1m. And of course you can do the same thing in backwards time. So if I reverse time, then I get the unstable horror sphere, which is this guy, and all of the normals to it determine geodesics that when I follow them backwards, so I guess I should draw the arrows the other way because I want to follow things backwards. When I follow these backwards, I reach eta, the point at which the geodesic of v originated. And so this gives the unstable leaf through v. So what did I do? I just defined two foliations of the unit tangent bundle. And let me work out what the dimension of everything is. Let me say the dimension of m is d. Then the dimension of the unit tangent bundle is 2d minus 1. The tangent bundle has dimension 2d, and I kill a dimension by restricting to unit length. Each horror sphere is a co-dimension 1 manifold in the universal cover. So the dimension of these horror spheres is d minus 1. The dimension of these guys then, which are the unit normal bundles over these guys. I hope everything adds up here. Well, no, it's the same. It's the same because once I determine the point on the horror sphere, the vector is uniquely determined. So there's no extra degrees of freedom by attaching these unit vectors. So the dimension of these guys is also d minus 1. So now what do I have? I have three foliations of T1m. First off, it's foliated by orbits, and that's one dimensional. Second, it's foliated by the stables, and that's d minus 1 dimensional. And third, it's foliated by the unstables, and that's also d minus 1 dimensional. And if I add them all up, I get 2d minus 1, which is the dimension of the manifold of T1m. So in other words, I've just produced all of the structure that you need for an Enosa flow. These are the stable manifolds, these are the unstable manifolds, and that's the flow direction. And if you take the derivatives of these, the differentials of these, then you get the bundles that you usually expect. So this is an Enosa flow. And for Enosa flows, this basic theory is all known. So let me make sure I say what I wish to say about it. So again, M has negative curvature, and phi is holder continuous. There's a unique equilibrium state. That equilibrium state is fully supported. By the way, let me emphasize that phi is a function on the unit tangent bundle. All of my pictures, I always draw the manifold itself, but phi is really defined on the unit tangent bundle, so it doesn't just depend on some point on the manifold, it also depends on the direction. What about this one? What should the geometric potential be here? Well, the geometric potential here, I'm not going to write a formula. I'm going to write the following. I'm going to write that this is the expansion rate of the, so phi, the geometric potential at v, because the dynamical system is on the unit tangent bundle, this is the expansion rate of the unstable horosphere through v. That is, if I want to know what the value of the geometric potential is at a point, sorry, it's negative the expansion rate. Then I look at the unstable horosphere. As I flow this horosphere forward in time, it gets bigger. And the rate at which it's getting bigger, the instantaneous rate at which it's getting bigger, is the geometric potential. And in particular, the integral of this is going to be the sum of the positive Lyapunov exponents of v. And then it remains true that the pressure of the geometric potential is equal to zero, and that the corresponding measure is the Louisville measure. The statement about analyticity of this function remains true. The statement about CLT and Bernoulli remains true. The statement about decay of correlations becomes harder because it's a flow. In some cases, one can say it's exponential, but it's a subtle question and I don't want to get into it. So this is the negative curvature case, and now in my last 13 minutes, let me say something about non-positive curvature so that hopefully I can at least formulate a theorem that is new and not simply classical. So now let me assume that my manifold is just as before, except that the curvature is allowed to take, the sectional curvature is allowed to take the value zero at some points. Well, you still have a universal cover, and in fact, you still have this whole picture. You still have stable horospheres. You still have unstable horospheres. You still have these filiations. The geometric picture goes through almost exactly as before. There are two differences. One difference is that in the uniform case, you know that the horospheres and therefore the stable and unstable manifolds vary holder continuously on V. You don't know that in non-positive curvature. As far as I know, that's an open question. It's not expected to be true, but there's no explicit counter example. Second, and somehow more severe, when you have negative curvature, these guys are transverse to each other. Now it doesn't look like they're transverse here, right? Those horospheres don't look transverse. The picture doesn't live on M. The picture lives in the unit-tension bundle. And when I say they're transverse, what I mean is that these guys meet to quadratic order, so that when you consider the normals, the normals separate, and the normals are derivative, so the normals separate at linear order, and that's the transversality. Now what can happen in non-positive curvature? Well, suppose I take this picture and let me make a change in the middle of it. Let me change the metric so that there's a strip here in which everything's just flat. This is the thing that you can do. What happens in the universal cover when I consider that flat strip? Suppose I consider a tangent vector v, which points exactly in the direction of that flat strip, so that its geodesic is a periodic orbit winding around the flat strip. So here's the geodesic of v. Well, the stable horosphere looks like this, and the unstable horosphere looks like this, and in fact they intersect. So that means that the corresponding foliations here are not transverse to each other. So this is the problem which arises in non-positive curvature. I mentioned the importance here that they separated at quadratic order. The problem here is not just that they coincide, the problem is really that they separate at slower than quadratic rate. So let me in fact make that a definition. Let me say a vector v is singular. If there are curves, let's say c1 and c2, on the stable and unstable horoscheres of v that separate slower than quadratically. This is a very informal statement, but as I'm running lower on time, I hope you will forgive me. You have two curves that separate to slower order than t squared. This would be a singular vector, but in fact you could make it singular by doing the following. Let me draw the same picture. Let me collapse that flat strip to a single point so that this part in the middle, in negative curvature, this would look like a surface of revolution of say 1 plus x squared or something like that. But if instead you make it 1 plus x to the fourth or something higher than quadratic, then you're going to get a single singular orbit, a single singular geodesic. And the definition that I need is that manifold m is called rank 1. If the singular set, that is the collection of all singular vectors, is not everything. As soon as you have a single tangent vector that is not singular, in that case we say it's regular, then we call this a rank 1 manifold. The reason for that terminology is not at all clear from the way I introduced it. The usual definition of singular vectors is given in terms of Jacobi fields, and once you give the definition that way, it's clear why this is the terminology, but I didn't want to write equations. I wanted to draw pictures. So we're interested in rank 1 manifolds of non-positive curvature. And in this setting, one can say something about the thermodynamics. One can say something about a theorem along these lines. The first result in this direction was given by Knieper, 1998 I believe, and he proved the following. He proved that if m is a rank 1 manifold of non-positive curvature, then there exists a unique measure of maximal entropy. So in other words, a unique equilibrium state for the zero potential. He also showed, let's call this mu naught, mu naught is fully supported. So every open set receives positive measure. Now there's a fact about this singular set which I didn't tell you yet. The singular set is closed, it's invariant, and it's nowhere dense. This is a fact that must be proved, but once you prove it, this result from Knieper immediately implies that the entropy of the singular set is smaller than the entropy of the whole system. This is because it's a closed invariant subset. If the entropies were the same, then there would be a measure of maximal entropy supported on the singular set. This is because the flow is entropy expansive, which I didn't say, but this is the way that the argument goes. So in other words he proved that the singular set, which is somehow the set of obstructions to hyperbolicity has smaller entropy than the whole system. Now if I start considering different potentials, that's not always going to happen. So for example, let me go back to my favorite one. If I consider the geometric potential and I ask what happens with its equilibrium states, I'm going to see exactly the same story as I saw for this Manville Po-Mo example back at the beginning. Let me think now just about this specific surface. I can take the periodic orbit measure supported on this single periodic orbit at which the curvature vanishes. Well the Lyapunov exponent there is zero. Because the Lyapunov exponent is how quickly the unstable Manville Horosphere expands. I erased the picture, but the point is for that example is zero. And the entropy is zero. It's a periodic orbit. So there are two ergodic equilibrium states in this example. There are two ergodic equilibrium states. There's one. It's not a delta measure, but you take my point. You have the periodic one, and you have the... And what does that happen? That happens precisely because the pressure of the geometric potential, which is equal to zero, is equal to the pressure of the geometric potential restricted to the singular set. In other words, this inequality fails. So in the last two minutes I can state a theorem. And the theorem is, somehow an analog of this, is that that is the only obstruction. Here's the theorem. If M is a rank one manifold of non-positive curvature, phi is Holder continuous potential function, or phi is a scalar multiple of the geometric potential. Remember the geometric potential, we don't know if it's Holder continuous. Our methods work for the geometric potential anyway, but it's a bit of a different proof. So you have a potential in this class. If the pressure on the whole system is bigger than the pressure on the singular set, then there exists a unique equilibrium state, and it's fully supported. On the other hand, if the pressure, if this condition fails, then there's an equilibrium state supported on the singular set. So this is an optimal condition. This is a dichotomy. Let me not say anything about that one. Ergodicity of the Louisville measure is a big open question in this area, and we are not going to approach it. Let me observe that, if your manifold is two-dimensional, then results of Ladrapier, Lima, and Sareeg tell you that the equilibrium state you get from this is in fact Bernoulli. Whether it satisfies some decay of correlations, whether it satisfies the central limit theorem, that's an open question. We expect that it does. I have some ideas how to prove it, but it's not there yet. And let me maybe just end by saying the following. One has, of course, to verify this condition. If I had no way of verifying this condition for some examples, then we didn't prove anything new. So I can observe one thing, which is that this condition is a C0 open condition. It's stable under uniform perturbations. But at the very least, we demonstrate that the class of potential functions for which you have the good thermodynamics is a C0 open set. So that's something. Because we know that gap holds for at least one potential by Knieper's result. As it stands, that's a little circuitous because Knieper proves uniqueness for this example. He proves it in a completely different way. His methods are very geometric, and so far as we can tell, there's no way to extend them unless you deduce this gap. From that gap, you get this gap as long as your potential function is very close to being constant. And from that, you get uniqueness. So it's a little circuitous. What we can prove is the following. We give a direct proof of this that does not rely on Knieper's paper. And the proof is such that if phi is constant on a neighborhood of the singular set, that gap holds. So it works not just for any constant potential but for a potential which is constant on a neighborhood of the singular set. Now, that's not so much. It's still fairly restrictive. But there are some cases where it's actually quite a lot. I will end by just saying something, which is that if I consider this particular example, the singular set is a single orbit. It's a single periodic orbit. In particular, it's uniquely ergodic. If the singular set is uniquely ergodic, then there is an open and dense set of potential functions which are cohomologous to something with that property. In other words, if the singular set is, I will write a couple of words, if the singular set is uniquely ergodic, then this condition holds generically. In other words, for the example that I drew, you get uniqueness for a C0 open and C0 dense set of holder-continuous potential functions. I don't know whether it works for a more general class of manifolds. That's sort of the next thing to do, but I'm a bit over time, so I'd better stop here. Thank you.