 Right, well, I'd like to start by thanking the organizers for the opportunity to come to this wonderful place. And so what I want to talk about is joint work with three friends, Vaughan Klayman-Hager, Todd Fisher, and Dan Thomson. And it builds on a program of Klayman-Hager and Thomson, which takes classical results of Bowen about construction of unique equilibrium states and does essentially the same job, but in a context where the ingredients that Bowen used aren't totally available. So I'll say a few things about that program towards the end of the talk. And if this talk encourages you to go and read their work, then I've done my job. And so roughly speaking, the talk is about sort of two pictures. There's a classical picture, which looks like that. And then there's the picture that we're going to get today, which looks like that. Some of you in the audience will already know what these pictures are. And well, so for some people, I will be saying some rather elementary things, but hopefully that will be OK. So let me start with something definitely elementary, the notion of topological entropy. So I'm going to work with flows, because I want to talk about the geodesic flow. So we have a flow. And then you have the notion of a Bowen ball. It's a very familiar notion. The idea is if I have a point x, and we flow along for time t, you're interested in all orbits, which stay within some epsilon of the orbit of x for the whole time t. So you look at all initial conditions such that you don't deviate by more than epsilon for time t. That's the Bowen ball. And they provide the open sets and some topology. And when you've got a collection of balls, you can try to see spanning means you want to cover the whole space with these balls. Separated means you want the balls to be pairwise disjoint. So the more balls you have, the easier it is to span. So it makes sense to count the maximum number of balls in a set that spans the whole space. Wait a minute. No, the smallest number of balls in a spanning set and the maximum number in a separated set. So I think by the time I get to the bottom line, it doesn't make any difference what number I put. So I think the motivation for epsilon to epsilon was, to epsilon separated means your epsilon balls look like that. And epsilon spanning means they look like that. And there's a, you can change from one to the other by switching, you either change epsilon to epsilon over two or two epsilon as appropriate. If I stand up here and try to do it, I will get it wrong. If you take a little bit of time and a sheet of paper, you can get it right. But anyway, if you think about, say, the separated things, this gives in some sense the maximum number of orbits of length t that you can distinguish if your eyesight is limited to resolution of epsilon. So if I take my glasses off, I have a somewhat larger epsilon. And then eventually at the end of this business, as you let epsilon tend to zero, you allow finer and finer resolution. And so the idea is that the number of distinguishable orbits in many systems will grow exponentially as the time you're looking at grows. And this topological entropy measures that growth rate. Okay, so that's topological entropy. And it actually came after measure entropy, which I'll describe now. And the description I'm going to give here is really due to Catoch, and it very closely parallels the construction of measure theoretic entropy, of topological entropy. The original definition of measure theoretic entropy involved a partition into measurable sets and was somewhat different. But so the idea is we do exactly the same things as we did before, except that the measure is involved because we don't try to span or cover everything. We're content to do some of the space. The number one-half is significant because it's a number between zero and one. You can pick any number between zero and one that you like and use that instead of one-half. So if you're thinking about the spanning sets, you want to cover at least a fixed positive measure. You don't try and cover the most difficult part. You cover the easy part. And so then you get a parallel definition of an entropy that depends on the measure. So this is the measure entropy for the measure mu. And then there is the variational principle. It's obvious from these definitions that the measure entropy is less than or equal to the topological entropy. And the variational principle says that the topological entropy is actually the supremum of these measure entropies over all ergodic measures. And so the definitions that I've given here are for an ergodic measure. You have to do something a little bit different if you want to think about an ergodic measure. OK, so that's entropy and measure entropy and the variational principle. The inequalities should go in the opposite directions. So if the second one is wrong, then the first one is wrong, too, I think. But if I'm covering, let's see, if I'm doing spanning, I want to cover, I think the second one should have been greater than or equal to a half. Sorry about that. So you want to cover at least a half, but you're allowed to pick the easiest half. And then with the separating things, you don't want to have too much. So I think I got those inequalities back to front. Yeah. OK, so let's see what happened. OK, so now I want to generalize this notion of entropy to the notion of pressure. So with pressure, we have a continuous function, real valued. It's called the potential function. And you make definitions that are very, very similar to the ones we made before to define entropy. So we have this sum over the elements of the, in this case, separated set. And for each element, we take the exponential of the integral of the potential along the orbit. And the simplest potential function you can imagine is to take phi equal to 0 everywhere. If you do that, this integral is 0, and e to the 0 is 1. And this sum is then counting the number of elements of the separated set. And so this is a generalization of the number that we looked at when we defined entropy. And what you're doing is, instead of giving each point in the separated set weight 1, you're giving each point a weight that depends on the potential function along the orbit. And it's the same deal if you use spanning sets. And then you go through the same limiting process to define pressure. And so if the potential was 0 here, you would end up with entropy. What you get now is the pressure of the potential phi. So that's topological pressure. And of course, there's now going to be measure theoretical pressure. So we have an ergodic probability measure. And I'm actually not going to write down the things we had before. I'm just going to write down what comes out at the end. You get the measure entropy, and then plus just the integral of the function with respect to the measure. So this is the pressure of the potential with respect to the measure. And the sort of fancy limiting stuff disappears into the entropy. And then you have the variational principle, the same as for entropy. So topological pressure is the supremum of the pressures over the ergodic measures. OK, so then there's a definition that a measure is an equilibrium state for the potential if the measure realizes the topological entropy. I think these measures are also called Gibbs states by other people. And these ideas came out of statistical physics and thermodynamics. So people like Ruel really understood the physics and moved the ideas into dynamical systems. People like me tried to read Ruel, and I confess I do not understand the physics. And I have no idea what pressure has to do, this pressure has to do with atmospheric pressure. If someone can enlighten me, I'll be very happy. OK, so equilibrium state. And now for reasonable systems. So my idea of reasonable is the infinity dynamics. You actually have, for any potential, you do have an equilibrium state. This is, I believe, first proved by Newhouse. I think there's a later proof by Sharon Broussille. So anyway, for the systems we're going to look at today, any continuous potential has an equilibrium state. But it might have multiple equilibrium states. A simple example would be if you took a diffeomorphism with zero topological entropy, then any invariant measure is going to have measure entropy zero, and it will be an equilibrium state. So unless your zero entropy system is uniquely ergodic, it will have multiple equilibrium states. And when you have a unique equilibrium state for a potential, then you've got something sort of interesting happening. You've got a reason to associate the measure to the potential. And so the hope is that if you look at interesting potentials, they will have unique equilibrium states. And these measures will be, therefore, interesting, and will hopefully tell you something about the system. So that's the sort of background philosophy. Right. So the context in which this kind of business works is when you have hyperbolic dynamics, something like an anosolphe diffeomorphism. For me, it's going to be an anosolphe flow. So that's why I'm going to start with a Romanian manifold with negative sectional curvatures. And the geodesic flow of this thing is the original example of hyperbolic dynamics. It's the prototypical anosolphe flow. So the geodesic flow, let me just remind everyone that the geodesic flow is not a flow on the manifold. It's a flow on tangent vectors to the manifold. And it's natural to restrict it to vectors of length 1. And if you have a unit vector on the manifold, it defines a unique geodesic, which is a curve with unit speed, which goes along on the manifold. It has the property that it gives the shortest path between nearby points on it. And if you follow the path for time t and look at the tangent vector after time t, that is what this is the image of the vector v under the geodesic flow for time t. So that's what the geodesic flow does. And there is a natural smooth measure on the unit tangent bundle, which is invariant under the geodesic flow. And so so far everything I've said is true for all geodesic flows, for all Romanian metrics, maybe even for all Finzler metrics. The fact that the geodesic flow is a nosov is a consequence of negative curvature. So let me just draw a picture of what the stable and unstable leaves of the look like in the unit tangent bundle. But I'll draw a picture, not of the stable and unstable leaves themselves, but of lifts of them to the unit tangent bundle of the universal cover. So if I have a surface of negative curvature, its universal cover is diffeomorphic to our rear. The prototypical example, which is imagine a surface of genus 2 with a hyperbolic metric. And then the universal cover is the Poincare disc. And the picture of a compact manifold with negative sectional curvatures being covered by its universal cover, pretty much looks like the Poincare disc covering a compact hyperbolic surface. Most of the features that you see in the hyperbolic picture with the constant curvature carry over to the general case. So if I have a geodesic in this picture, it joins two points on the boundary. And the stable manifold for a vector tangent to this geodesic consists of tangent vectors of geodesics that head to the same point on the boundary. And there is what's called a horosphere. And if we were in the case of a surface, it would be a horocycle because then the curve would be a circle. And we look at the unit normals to this horocycle or horosphere that point inwards. And these geodesics flow off. And if you do some calculations, you can show that these geodesics approach each other exponentially in forward time. So if that was v, this is the stable manifold for v. And I've made a construction in the universal cover. You can just project this down to the compact manifold or to the unit tangent bundle of it. And if I'd made this construction using a different lift of the same vector from down below, what I'd get would just be the image of this picture under an element of the covering group. And then the unstable manifold is the same thing except that you use the other end of the geodesic. And you use the output pointing normals because the vector v points out of this horosphere. And so that will give w u v. And these geodesics approach each other as you go backwards in time. So that's a sketch of the sort of hyperbolic structure for the geodesic flow and negative curve feature. And maybe let me just remind you that in the case of the Poincare disk, these horospheres are exactly in the Poincare disk model. These horospheres are Euclidean spheres that are tangent to the boundary. OK, so you have this hyperbolic structure. And now Bowen showed that nice potentials, holder continuous potentials, always have unique equilibrium states for this Anosov geodesic flow. So it's really a result about Anosov geodesic flows. And what drives the proof is the hyperbolicity, the uniform hyperbolicity of the system. OK, now the most interesting potential to study is what's apparently is now called the geometric potential. It might be that I should have put a minus sign into my definition. Since I didn't, I will later look at negative multiples of this potential. Other people put the minus sign in here and look at positive multiples. So what is phi sub u? Well, the idea is you look at the unstable manifold. And as it flows forward, it expands. And you measure the rate at which that is expanding. And so, of course, this is the infinitesimal expansion of just this is the infinitesimal expansion, rate of expansion of volume in the unstable direction. And so this is clearly both dynamically and geometrically important quantity that's very relevant to the behavior of the flow. OK, and so the picture that I drew here, I can now let me label the horizontal axis. That's some real number Q. And well, in this direction, you have the pressure. So this is a graph of the pressure, the topological pressure of minus Q times this geometric potential. And this is in the case of negative curve feature. So when Q is equal to 0, you get the topological entropy of the geodesic flow because pressure reduces to entropy when the potential is 0. So in the case of surfaces, this is a topological quantity, something to do with the genus of the surface. In higher dimensions, I think you get what you get. And then we cross the axis here at 1. And the reason why we get 1 here is because, well, let me explain that in a minute. So what was that? Well, let me just, oh, I see. Yeah, so it's basically the sum of the positive exponents. And then in the surface case, it will be the, wait a minute. No, sorry, you get this gives a function. It tells you how things are growing. So in the case of a surface, it tells you how things are growing in the unstable direction at each time. If you then take the book off average of this function, then you get the positive exponent. On the case of a surface, you get the positive exponent. In higher dimensions, you get the sum of the positive exponents. Oh, so this norm, so this is some sort of determinant. Yeah, I should probably have called it determinant. So you look at, you've got a linear map. And you've got a notion of volume. And you're seeing how the volume is growing. Yeah, norm probably wasn't a very good notation. Sorry about that. Yeah, and so the point of this, one of the points of this phi sub u is that when you form its book off average, you get the Lyapunov exponent, or the sum of the Lyapunov exponents in the unstable direction. And actually, there are two different functions that you could use for phi sub u in the geodesic flow context. You could either look at the picture in the unit tangent bundle, or you could look at the picture in the manifold and measure volume in the horosphere. You get two, I think the correct word is cohomologous. The two functions, after you've formed the book off averages, either way, you get the same book off averages that are the sum of the Lyapunov exponents. So we have this potential. And now, the main feature of this picture is that this curve is convex, or if you look at it the other way, concave, it bends that way. And the reason for the structure of the curve is the variational principle. So there is a picture that I can draw with a computer. And it shows what happens if you fix the metanergodic measure mu and look at the pressure of this quantity for different values of q. And because you just get the entropy for the measure minus q times the integral, you get a linear function. And so for each ergodic measure mu that you can think of, there is a straight line in this picture. And the lines have, they intersect here above the axis because the entropy is at least 0. They have negative or 0 slope because the integral of vu is positive. And because of the variational principle, this curve that I've got is the envelope of all of the lines. So at each point there is, so you have lines. And because of Bowen's result, for each q, each point on the graph, there is a unique equilibrium state. And that gives a line that is tangent to the graph at that point. So this curve is actually smooth. They did much better than smooth. It's even real analytic. But at each point, there is a tangent line which corresponds to the equilibrium state for that multiple of vu. And so the measure that we get at this point is the measure of maximal entropy. That's the measure that was constructed by Margulis or by Bowen. In a geometrical context, there's a construction of Patterson and Sullivan. So we get a very important measure here. The measure we get at this point is the Liouville measure. That's the natural, smooth, invariant measure. And the reason why Liouville measure works is because this integral that we've got gives the sum of the positive exponents for the flow. And when you have a measure that's absolutely continuous with respect to volume, Pessin's formula tells you the entropy is the sum of the exponents. So then we get 0. So that's why we get, by the graph, crosses the axis at Q equals 1. And the tangent line there has a slope that's, well, the tangent line there corresponds to the Liouville measure. So that's the classical picture. And now I want to move on to something a bit less classical. We're going to think about a surface. And maybe one thing I should say is that this picture is a picture for the case of a surface. Unfortunately, oh, no, you can draw this picture in general. But there's something else you can do. There's something that you can do in the surface case that you cannot do in higher dimensions. So in the surface case, if I think about some value of Q and I look at the tangent line, I get an entropy and I get some number there. And these numbers are meaningful. So the tangent line is given by the equilibrium state. This number here gives the Helfstorff dimension of a certain set. It's the set of points of things in the unit tangent bundle, which have Q, which have, oh, so sorry, there are three numbers. There's also the slope of the line. And if you look at the vectors of the unit tangent bundle in which you see the Lyapunov exponent that gives the slope of this line, then this number delta is a Helfstorff dimension related to that set. What you actually have to do is look in the stable manifold and then look at the subset of that where this Lyapunov exponent is realized. Then that subset of the unstable manifold has Helfstorff dimension delta. And then this is the entropy on that set. And this tie up between Helfstorff dimension and exponent and entropy is a two-dimensional frame. Life is just more complicated in higher dimensions. And so now that I come to non-positive curvature, I'm going to restrict attention to surfaces. It's also partly because the results that we have are fairly complete in the case of surfaces and rather incomplete in the case of higher dimensions. So for surfaces, you again have the geodesic flow. And now the picture that I drew here of what can happen in, the picture looks almost the same, except there's a difference. So one thing that I probably should have emphasized when I talked about the negative curvature case is that the stable and unstable leaves are transverse to one another in the unit tangent bundle. If you look at this picture, it looks as though they're tangent to each other there. But that's deceiving. They're only tangent after you have projected down to the manifold. Up in the manifold, these things are not tangent. And the reason they're not tangent is because if you look at the second all, these things make first-order tangency. At the second-order level, they are not. You do not have second-order contact. The second-fundamental forms of these things are different. So in the surface case, you're just talking about the curvature of these two horror cycles. And so in the case of negative curvature, both of these horror cycles have non-zero curvature. One of them curves one way. The other one curves the other way. And in non-positive curvature, you still have the two horror cycles, they're tangent to each other. And now they have curvatures that are greater than or equal to zero if you look in the right direction. But one of them bends one way and one the other way. And so they can make second-order contact with each other. And so with the regular geodesics, that's when you've got the sort of behavior you see in this picture. The singular ones are where you see second-order contact. So the simplest example, the clearest example of second-order contact is if you think about a surface of genus 2, which has negative curvature except for a flat cylinder in the middle. And if you look at the geodesics that go around the flat cylinder, these geodesics are just parallel to each other. And if you look at the horror cycle for one of these geodesics, the horror cycles in both directions are just straight. And the geodesics perpendicular to them are lifts of these parallel circles. So you get parallel lines in the universal cover. And so the universal cover has a flat strip in it. You can also get an infinitesimal version of this where you just have one geodesic in the middle, along which you have zero curvature at all times. And you get an infinitesimal flat strip. So in the case of a surface, there's a very simple criterion to decide whether a geodesic is regular or singular. If at any time the geodesic goes through negative curvature, it's a regular geodesic. If it's sometime, oh, sorry, and if it never goes through negative curvature, that means you've got zero curvature at all times along the geodesic. That's when you get a singular geodesic. There's a corresponding criterion to distinguish the two in higher dimensions, but it's a little more complicated. But the idea is that for the geodesic to be singular, you have to have zero curvature for all along the geodesic. But in higher dimensions, it has to be organized in the right kind of way. And I'll not say more about that. So the unit tangent bundle is the union of regular and singular, because basically singular means not regular. And the regular set is open. That's obvious from the definition. And it's dense. That's very easy to prove in the case of surfaces. In higher dimensions, it's the theorem of Werner Bauman. The geodesic flow is ergodic on the regular set. That means ergodic with respect to the Leeuville measure. For surfaces, it's a theorem of Yashapesian, who is in the audience. And in higher dimensions, it's a theorem. It's work of Yashapesian together with results of Werner Bauman and Misha Brinn. And I also did the same part of the story that they did. So unfortunately, we do not know that the geodesic flow with respect to Leeuville measure is ergodic on the whole of the unit tangent bundle. Because we don't know what the Leeuville measure of the regular set is. That is an open problem. There was a period during the 1980s when we all believed that the answer to this question is zero. Unfortunately, the argument which was supposed to show that the measure is zero just doesn't work. And since then, it's an open problem. The only known examples in the case of surfaces of geodesics, of singular geodesics, look like this. They're closed geodesics that either bound the flat strip or give an infinitesimal flat strip. We do know an example of a singular geodesic and a c-infinity surface of non-positive curvature other than closed geodesics. On the other hand, there is no proof that this set has measures zero. Okay, there is a measure of maximal entropy for a compact manifold of non-positive curvature of rank one. So in the case of surfaces, anything of dimension greater than a record, of genus greater than a record two. This measure was constructed by Gerhard Schneeper and he basically used the Patterson-Sullivan construction. He also showed that you can construct the measure as a limit of measures supported on regular closed orbits. Now, in the case of a surface, it's easy to see that the entropy on the singular set is zero because you've got zero curvature everywhere along the singular geodesics and that leads to Lyapunov exponent zero. There is just no hyperbolicity whatsoever on the singular set for surfaces. In higher dimensions, you can have hyperbolicity on the singular set, but Schneeper showed that the entropy on the singular set is strictly smaller than the entropy for the whole flow. Okay, so now I want to draw the picture for the case of a surface of non-positive curvature that corresponds to this one. And so I'm going to assume that we have a surface, we have curvature less than or equal to zero, and I'll draw the picture in the case where the singular set is not equal to the empty set. If the singular set is equal to the empty set, we've essentially got this picture. We have a nosov geodesic flow again. And so you get the main point of the talk is that you get the same picture with just one difference. Let me describe the difference first. This is at the point where q is equal to one and you definitely get a corner in the graph because there are two measures you can think about. One of them is the Liouville measure, or actually the restriction of the Liouville measure to the regular set. That is an equilibrium state for... So we're still looking at this. This is an equilibrium state for minus phi u for the same reasons as here. That gives the tangent line on one side. On the other hand, if you pick a measure that's supported on the singular set, just take, say, a measure that's on a single closed orbit with zero curvature all along it. The entropy is zero, the exponent is zero. You just get the horizontal line, but it goes through the same point. So you get a horizontal line and also a line with nonzero slope. So that gives the corner here. And up here you have the measure of maximal entropy again. So that is the... And this curve is certainly convex for the same reasons as before. So what the four of us are able to show is that this is the only corner in the graph and that at all the other points on the graph you have, in fact, a unique tangent line. And that's because you actually have a unique equilibrium state. And also out here you may not have unique equilibrium states, but you do have the horizontal tangent line, that the graph is the horizontal line there. So... What was that? We haven't actually proved that. It should be. Vaughan is very confident that he is. And I believe Vaughan. So he will... So it should be. But we haven't actually done that yet. But we do know that we have unique equilibrium states. So those are the main results. So for the geometric potential, if we take multiples of it by Q less than one, we get a unique equilibrium state. And if we have a holder continuous potential and the reason for this interval restriction on the values of the potential is to make sure that if we start having a potential that's larger than the topological entropy, we might be able to get non-unique equilibrium states in the singular set by having the integral term to dominate the initial entropy instead of the entropy term. Okay, so I now want to give a very brief sketch of how we go about proving this. And there's a sort of iceberg. I've been describing the stuff on top of the iceberg. I now want to allude to what's under the water. And this is this program of Kleinman, Hager and Thompson. And their idea is to take the classical result of Bowen and make it work in places where Bowen couldn't make it work. So Bowen says that you get a unique equilibrium state if you have expansiveness, specification, and the Bowen property for the potential. So the Bowen property of the potential says that whenever you have two orbits, two orbit segments that stay within epsilon of each other for the whole time between zero and t, then the integrals of the potential along the two things are close to one another, or at least uniformly not too far apart. Expensive means that if I have two orbit segments, or say two orbits, that stay close to each other for all time, close enough to each other for all time, then the picture doesn't look like what I drew. It just looks like two points on the same orbit. And so if you look at the picture of the surface with the flat cylinder in it over there, you can see that as soon as you've got a flat cylinder, the geodesic flow is not expansive because you can just have two orbits of the geodesic flow that go around and around in parallel and stay at a constant distance from one another. And then specification is very complicated to describe if you write out the formula. It's reasonably clear if you draw the picture and don't worry too much about the details. The idea is that you have some pieces of orbit that are prescribed, and then you want to find a single orbit that follows each of these pieces in succession. So you want to start off and follow this one very closely. And then you want to follow this one next. Well, you have to have a certain amount of transition time, but you have a uniform upper bound on your transition time. Then you follow the next piece very closely. Then you take some transition time, but not too much. Then you follow the next piece and so on. That roughly is specification. There's a closely related notion of shadowing. What's different in shadowing is that there are no transition times and the pieces of orbit that you are following just have a small jump. So it's reasonable that you can follow a sequence of orbit segments like that. If you have these gaps, then there has to be a transition time. And so specification says that if you have pieces of orbit and allow yourself a large enough transition time, then you can do this sort of thing. And if you have all three ingredients, Bowen can produce a unique measure of maximal entropy. And the idea of how you do it is roughly you start with a collection of orbit segments that gives a good large sum when you're trying to create the pressure. And then you look at the distribute measure along those orbit segments. And then you hook them up with this construction and you can make it into a closed orbit. And then you take a limit of those things. And then to prove that it's unique, you imagine that if you had two measures of maximal entropy or maximal pressure, you can sort of hook the two things together using specification and build something with even larger pressure. Something kind of like that. Bowen's argument is clearer. You can do it if you have all of these ingredients. If you don't have these ingredients, what do you do? Well, you try to restrict yourself to the places where you do have the ingredients and ignore the places where things don't work. So let's think about orbit segments. So the idea is you want to think about an initial point x and a time t, which is how long the orbit segment goes for. So you have point x and you go for time t. So the convention is x is at the beginning of the segment. And star is just concatenation. And now you think of three subsets. To be honest, you really only need two subsets. G stands for good. P is meant to be P is prefix. S is suffix. P and S are essentially bad. So the idea is if you have any orbit segment, you want to be able to chop off the beginning and get something in the prefix set, chop off something at the end from the suffix set. And what's in the middle is good. And then the idea is that good orbit segments will have expansivity and specification. You have the Bowen property for good, for the potential on good segments. And if you try to do pressure but only use Bowen balls coming from orbits that come from the prefix or the suffix set, then you don't get all of the pressure. And then if you... So that last property says that the sort of bad stuff of the beginnings and the ends really doesn't make any difference. And that all of the action is really coming from the good orbits where you can do the sort of things but Bowen did. And then Clayman Hager and Thompson have carried out this program initially for various symbolic systems. And laterally, they have a version of the result for flows, which is behind the result I'm describing today. So that paper, all of those papers are available on Archive. So, yeah, I encourage you to read them. So let me just say something very brief about how we make the decomposition. The idea is that we want this middle part to have some hyperbolic behavior. And we see hyperbolicity in our geodesic flow when the two horror cycles both bend away from the normal direction to the geodesic. So to be bad means that this bending away is weak and that that happened and that it's weak too often. So there's a definition of being a bad up there. And then the way we do the decomposition is we just start at the beginning of the segment, take the longest initial segment that's bad, get rid of it, then go to the other end, take the longest tail segment of what's left, which is bad, throw that away. Then what's left in the middle is good. And you get hyperbolicity properties for these good segments that allow you to do the things you want to do. And we can more or less copy the usual argument for proving order continuity of things to show that we get the Bowen property for the geometric potential on the good orbits. So let me finish there. Thank you.