 Čakaj, Ali. Čakaj, organizatora, začal mi, da bi se počutiti. To je zelo način, če njih ne zelo počutiti. Čakaj, zelo počutiti. Čakaj, zelo počutiti. Zelo počutiti o entropiji, vsega vsega vsega vsega vsega. tako, da bomo kštati, in zanim sem odpočen, načo se vse bo povoljno odmah z vse. Tudi sem priznačen, da sem morače našli na vse, da močimo zašli na tebe, na mnoj zvrst, m je kompakto vse, ker so v svojo vzve, da spokojno vse, je zvrstva, so je kompakto vse, z mte vse. To je CR-Decal. Vse pravno vse. Kaj je napravil entropij, napravil metrič, vse je vse. Dostanje v 2 počke. Vse je napravil n, vse je napravil vse. Dostanje vse. FI of X, FI of Y, for zero, smaller equal, i smaller equal, n. And then with this metric, try to cover the space with balls, and you look at the minimum number of balls which you need. And this should be, let's say this is A of, it depends on f and then epsilon greater than zero. This is a minimum of the cardinality of the sets s, such that s is an epsilon generator, which means that the union of the balls with this metric, center x and radius epsilon, where x is in s, covers the entire space. And then you look at the exponential rate of growth of this. So let me give the definition, is limit epsilon goes to zero, but limit n goes to infinity, one over n log of this number. This is a topological entry. On the other hand, there is a metric entropy, which I will talk about the metric entropy, too. I suppose that I am in the same sitting, but now I will suppose that I also have an invariant measure, which is a probability. This means that the push forward is the same for f. And then one can define the metric entropy of f with respect to mu. Well, I won't give the definition. Basically, it's a similar definition, but now you ask that this set s, this balls cover almost all the space with respect to the measure. These are some definitions. There are many other equivalent definitions. It can be done with open sets, with other type of numbers here. For the metric entropy, it can be done also with partitions. These are the entropies I will talk about. They are well studied in dynamical systems. There are some important invariants. This is a topological invariant. This is invariant under major theoretical conjugacies. They describe somehow the complexity, the level of the complexity of the orbits of the system. They are also related with other invariants, like the opponent, volume growth, dimensions. So they are well studied and they are very important in dynamical systems. Now, more specifically, what I'll look at. I'll look at these as functions for the topological entropy. This, of course, can be seen as a function from some space of the geomorphism, where a function goes to the entropy of the function. And the metric entropy also can be seen as a function, but it will be defined on the space of invariant measures for some systems. Now, I suppose that M of F is the set of invariant measures. And then I look at the function. It goes from M of F to R. Well, actually R plus is always greater or equal to zero, where a measure goes to metric entropy of F with respect to this measure. This is called also the entropy function. And this is, I can show that it is an affine function. And well, what I'll look, more specifically, I'll look at the continuity properties of this. So, here I have a topology depending on R, depending on the space I take. Here I have, for example, the weak start topology set of measures. Now, some general things about these functions. Let me start with the second one, metric and H mu of F. As I said, this is an affine function, in general is not continuous. And what usually fails is general not continuous. And in many cases is not lower semi-continuous. This is generically. That's because, for example, some invariant measure can be approximated by periodic orbits. Measures supported on periodic orbits. The metric entropy on a periodic orbit is zero. And the metric entropy on another measure can be non-zero. So, this usually fails. However, in many cases, and there is a lot of work in smooth ergodic theory, I don't know, when this function is upper semi-continuous. Many cases, upper semi-continuous. And this is useful because this, well, is related to thermodynamic formalism. This will imply that there exist equilibrium states. In particular, measures of maximum entropy, which are measures where this metric entropy coincides with the topological entropy. I didn't say, but there is also a variational principle, which says that this topological entropy is the supremum of all the metric entropies for all the invariant measures. Okay. And there are many cases where this works. I want to say first, no? Maybe I should say what is the main tool here to prove this thing from Bowen. I don't know, it was made better by other people. It's something which is called entropy expansiveness. Well, there are generalizations, like asymptotic entropy expansiveness and other types of generalizations. This basically means that one can look at these balls, but when n goes to infinity, and what one wants is that there is no entropy in these balls. Dynamical balls, they are called dynamical balls, but there is very small entropy on these balls. And this usually implies that this function is upper semi-continuous, and this gives equilibrium states and measures of maximum entropy. Okay. Now, some general things about the topological entropy. And again, I will talk about upper semi-continuity and lower semi-continuity separately. For upper semi-continuity, there is one obstruction. Well, actually there are two, but for two together, let's say, is finite regularity, low or finite regularity, homoclinic tangency. What is a homoclinic tangency? There is a hyperbolic point. This will have a stable and a unstable manifold, and these two manifolds are tent at some point. And here there are some examples, explicit examples by... I should mention, I mean, Mr. Revich. That's when you have this, and you look at the entropy function, well, the topological entropy function, upper semi-continuous. This is for any smaller than infinity. This is what I mean by finite regularity. In a space of CR defiomorphism, with R finite, if there is this homoclinic tangency, at this point you won't have upper semi-continuity. And maybe I should say here that this... Well, you can... With this you can construct an example where you don't have the upper semi-continuity of the entropy function, too. So, in general, goes to H2 of F. Upper semi-continuous. It's equivalent with having the... This function goes to... Let me put F0. Goes to H of F, upper semi-continuous at F0. The upper semi-continuity of the metric entropy function and the upper semi-continuity of the topological entropy are very related. And the same tool, for example, the entropy expansive net proves the two. Now, as I said, there are these examples where this doesn't happen. And actually these are the only obstructions because there are two theorems. There is... I think he did it for the topological entropy, I guess. Newhouse did it for the metric entropy function. So, they proved that in the C infinity topology they are absolutely continuous. So, this doesn't happen. How should I put it? This infinity, upper semi-continuous. And if F is C infinity to H of F is upper semi-continuous. So, in the C infinity topology everything is fine. Now, also there is a result which says that far from this homoclinic tangency everything is fine, too. So, this is a result by... It says that the coefficient that you want topology here, this means that it's away from homoclinic tangencies. This is a set of homoclinic tangencies, the function has homoclinic tangencies, and the closure of it. The topological entropy is upper semi-continuous. F is here. And the metric entropy function is also upper semi-continuous. So, this gives a pretty complete description, I think, about the upper semi-continuity. Well, of course there are many other results for some functions with some specific properties. One can get some... the upper continuity of the metric entropy function, too. About the lower semi-continuity. Here, there are some results in low dimensions. And the most famous result is probably the result by cut-top, which says that if M is a surface, if we look at the space of C1 plus alpha, then on this space the topological entropy function is lower semi-continuous. It's... You can do it in the C1 topology, and you can put the restriction that the function is C1 plus alpha, too. OK. Well, if the dimension is one, then, of course, the entropy is constant zero, so it's obvious. And let me put, like, an observation. If I mention greater or equal than three, then you can look even at the C infinity functions, and this will not be lower semi-continuous. So these are some general facts about this entropy view that functions of the system or of the measure. Now I want to look at the case of partial hyperbolic systems. Let me define quickly what a partial hyperbolic system is. I'll denote it as PHD, partial hyperbolic defiomorphis. If there exists a splitting into three sub bundles, sub bundles are invariant, their star can be... And, well, three conditions, let's say, F, restricted to ES, is contracting uniformly with a stable space. On the unstable space, on EU, is expanding uniformly. In the center is... Let me just say that it is in between. This means that it can be contracted or expanded but weaker than what is happening on this other sub bundle. Why do I look at these systems? The reason is that, because for these systems, usually you can see lots of things. So the first thing is the center zero, then F is uniformly hyperbolic. And in this case, there are lots of things known. For example, it is known that they are entropy expensive. So the upper semi-continuity comes immediately. And, well, it's even more. They are structurally stable. They are conjugated to small c1 perturbations, which means that the entropy, the topological entropy, is actually locally constant. This, everything is very fine. Now, let's look if the dimension of the center is slightly bigger, equal to one. What is happening in this case? Well, one can still expect some good behavior, because, well, in dimension one, the entropy is fine, and the other directions are uniformly hyperbolic, so everything should be fine. And here, I just have, it is enough for me to say that the partially hyperbolic defilmorphism with the central dimension one are away from tangences. I will denote hd1. So these are inside ones which are away from tangences. So, I can just apply the result by Ljavo, Vienna, and Young, and the upper semi-continuity comes for free, for the metric entropy function and for the topological entropy. Now, about the lower semi-continuity of the topological entropy, this is a bit more complicated. So how about the lower semi-continuity of the topological entropy? Here, there is a conjecture, I believe that it's still open, that in this space the topological entropy is also lower semi-continuous, which implies that it is continuous. This is known to be true, but this can be proven in case-by-case analysis. One can show it for basically all the examples. Unfortunately, there is still no good classification of this, even in dimension 3, as far as I know. See, there are, lately, there are many results in this direction. Also there are new examples which keep showing up. Let me just say that in many examples, actually this is locally constant, and I think the most difficult case was perturbations, time one of hyperbolic flows. In this case it's actually continuous, it's not constant, it's easy to see. Well, there are several results here, I could mention, for the most general, it's myself with perturbations, including perturbations. Well, let's say the proof, in the proof you have to argue that it's close enough to the hyperbolic flows. For large perturbations, I believe it's true, but it's... So it is true for small enough perturbations, maybe I should say. This is the proof. C1 small enough perturbations of hyperbolic flows. We needed that it's close, C1 close enough in order to preserve some of the structures which exist for the flow and use them in the proof. That's locally constant, I'm not sure about some last examples by... I didn't check, if there is any difficulty. The last examples by Bonati with Bob Golev, and Po3. But I think probably the same technique would work. They have basically perturbations, some perturbations of the flows. Well, unfortunately as I said, it would be nice to have a proof in the general case, but as far as I know, there is no such proof. And let me talk a bit about the higher dimension of the center that is equal to then... Well, everything fails. In general, it's upper semi-continuity and no lower semi-continuity. Well, no upper semi-continuity, it is not see infinity. In the center you can have a homoklinic tangency, and there is no lower semi-continuity, even in the see infinity topology. You can construct example where it is not lower semi-continuous. One can... One wants to prove something, he has to add some extra conditions here. Now, how about the flows? Well, what is a partially hyperbolic flow for me? Partially... Well, I didn't say what's the entropy for a flow, but yeah, maybe you can use that. You probably get the upper semi-continuity at the measure of the maximum entropy as close to the measure of the maximum entropy. But it still depends, wait. If it's low entropy, but it depends if it contributes or not, total entropy. Because it's... If you have a tangency, I don't know, it depends. You have to put some conditions there. I think that if you put the right conditions, probably you can get the continuity. But for example, a homoklinic tangency product with an asshole doesn't work. And the homoklinic tangency can be a small perturbation of the identity of a small entropy. Ok, for flows, well, I didn't say, but the topological entropy is defined as a topological entropy of the time one map of the flow and the magic entropy is the same. The invariant measures for phi and for phi one are not exactly the same and one has to be more careful here, but basically one defines them like this. Now, for me, a flow is partially hyperbolic. I will denote this partially hyperbolic flow. If, again, I have the invariant splitting, and they are invariant under the flow. And again, it will be the same. Yes, it's contracting. It's exactly the same definition. U expanding and E c is in between. Let me just put an observation here. Or maybe I should put it in definition. What I mean here? Well, I want to put a restriction here and I'll assume that phi has no fixed points. No singularities. Probably some things can be done for singularities, but it's easier if you assume that there are no singularities. And let me just make an observation that the center bundle always contains the direction of the flow. So it will be at least one dimension in this case. But the one-dimensional case is the uniformly hyperbolic flow, is uniformly hyperbolic. In this case, again, everything works fine. It's entropy expensive, so upper semi-continuity comes. And it's, well, structurally stable modulus, time change, so the entropy is, in fact, continuous. This is everything. So, well, the question is what happened in the central dimension is equal to 2. Now, here I should say, like a general observation, is that the flows in dimension n are basically at something in between the show morphisms in dimension n minus 1 and dimension n minus 2. There are some restrictions of the show morphisms in dimension n, but some generalizations of the show morphisms in dimension n minus 1. So this is, let's see, that it's morally in between partially hyperbolic the show morphisms with the center 1, partially hyperbolic the show morphisms with the center 2 dimension. So one would expect that the behavior should, some results which work for this could be transferred for the flows. But possibly not all of them. And this is exactly the case here. On one hand, let me put it like a theorem, is, well, we still have to write it down, the upper semi continuity works, partially hyperbolic flows 2, this means that the center dimension is 2, topological entropy, is upper semi continuous, here in the C1 topology. And the metric entropy again is upper semi continuous, upper semi continuous. Of is, it's not immediate generalization of this of the show morphism case, but it's almost. So apply the same method. Because it is entropy expensive. And it's entropy expensive. So how about the lower semi continuity of the topological entropy? Here is where the difference comes. The topological entropy is not lower semi continuous. Not even in the C infinity topology. For this, we have an example. And the example, maybe I should give an outline of it. How much time do I have? The example is to take hyperbolic flow. Let's see. Hyperbolic, uniformly hyperbolic. On some manifold n, it can be, I don't know, the unit tangent bundle surface was negative curvature, suspension flow of an asshole. Then to take m, the product of T1n. To take a function, infinity, let's say, non constant. And to define the flow on m, x is y, x is in T1, y is in n. You keep the horizontal fiber fixed, and then inside the second coordinate you move with the speed alpha x. X, T, y. Move on the flow, but with different speeds. Some points you move faster, on other points you move slower. Then the topological entropy of this, the topological entropy of psi multiplied by the maximum, the exponential, maximum of alpha, the exponential of maximum of alpha. And how to perturb it, you make the flow move a bit up. The idea is, let's say that this is generated by x, some vector field x on n, then phi will be generated by the vector field zero x. So you can make an arbitrary small perturbation, c infinity small, phi epsilon to be generated epsilon d dx. And then the orbits will move up, and they, well, they go around the circle, and it's easy to show that the entropy of phi epsilon is the entropy of psi multiplied with the integral of alpha for log of the, I think it's log of the integral of e to the alpha. It drops immediately and it stays constant actually for every epsilon. Let me finish with just saying that, okay, it doesn't work, but at least there is still some hope one can, at least in some cases one can prove something, but I don't want to, let me just say that one can define something which is called the volume growth. Maybe you should, perfect. Let me stop that.