 Thank you for your invitation. And I'm very happy to be here at the so beautiful place and present our work here in this conference. The title is Upper Semi-Continuity of IntrifyMap for No Uniformly Hyperbolic Systems. This is a joint work with Gong Liao and Shi Rou Wang. Let's consider the topological system, meaning the continual map on a compact manifold, compact metric space, and consider all the F invariant probability measures, which is not a input. The IntrifyMap is this map. Given a measure, a point is a measure theoretical entropy. So this is a map. And as we already know, measure a theoretical entropy measure dynamical complexity. And it is an invariant for its morphic probability systems. And this IntrifyMap is not continuous in general. For example, let's consider the hyperbolic toroid automorphism. In this example, we take an ergodic measure with positive entropy. For example, the entropy is a larger than zero. Then this measure is approximated by a theoretical orbit atomic measures. This is a no result, for example, from Sigmund. So clearly the theoretical measure has no entropy. And this measure has positive entropy. So the picture is like this. Here is a measure with positive entropy. And this measure is a theoretical measure. And in this measure, we have a positive entropy. And this measure is the entropy is zero. So from this picture, you see easily this map is not continuous, of course. It's not lower semi-continuous. So if we say continuous for IntrifyMap, we mean upper semi-continuous. We already know the following facts. Expensive homomorphism of compact metric space for this space, for this topological systems, we have the upper semi-continuous. And for ExpensiveMap, we have this property. And also for a semi-particle entropy ExpensiveMap, we have this property. And also for C1 diffuses far from tendency, we have this map, we have this property. And also for C-infinityMap, we have this property. Now we consider the property for no uniform hyperbolic system. The definition of no uniform hyperbolicity, we take this definition. Well, M is a compact many manifold. And we consider one C1 diffule on the manifold. And then we take three numbers. Lambda S, lambda U, and the Epsilon. Epsilon is quite smaller than this two quantity. And then for each K, we define lambda K. This one depends on these three parameters. To be all the point X for which the following truth, there is a splitting with the invariant property. This is the invariant property. And satisfies the following. The first one, if we consider the restriction to ES bundle, then this normal is less than this quantity. And if we consider the restriction to the EU bundle, bundle, then this inequality holds. And also the angle between the two bundles decreasing not very fast, not very fast. Then we consider the union of lambda K and call this side passing side. And in this case, we call lambda K a passing block. The picture is like this. Here we have one block. And here we have another block, lambda two, and so on. In each block, the splitting is continuous. The splitting is continuous, and it is uniformly hyperbolic. But each block is not invariant in general. If we consider the union, it is an invariant side, but it's not compact in general. Now we consider all the invariant measures are put on the passing side. That means, sorry, a measure belongs to this side if and only if lambda has a measure one. And the invariant measure with no zero reapplicant components is called a hyperbolic measure. For an ergodic hyperbolic measure, we could define a passing side associated to it in the following way. Let omega be the oscillatory basin of new where all reapplicant components exist by oscillatory theorem. And this one has a new measure one. Do not es and eu direct some of the oscillatory splitting with respect to negative and positive reapplicant components respectively. And let lambda s be the norm of the largest reapplicant components of vector field, vector in es and lambda mu, the smallest one in eu. And consider epsilon much less than this. So this means for an ergodic measure, the reapplicant components are constants. These are positive reapplicant components and these are negative reapplicant components. So we consider this guy. And in this case, omega is a content in the passing side we defined previously. Now we denote epsilon bar, which is a decreasing sequence of real positive real numbers, which approach zero. And denote this sub side, this is the sub side of this side of measures with this property, with this property. This means what? This means the measure could not have a very large side with a large measure in the lambda k. We are k is very large, very big. We call this epsilon bar hyperbolic rate and Newhouse proved the following. He proved that the entropy map, if we restrict to this sub side, then it is upper semi-continuous. Roughly saying for one plus half, no uniformly hyperbolic defuse, the entropy map, when restricted on the side of hyperbolic measures with the same hyperbolic rate, that means the epsilon bar. Then in this case is upper semi-continuous. Question, is the entropy map upper semi-continuous when restricted to hyperbolic measures without the same hyperbolic rate? Okay. Yes, this is a measure supported, sorry, this is lambda. Yes, yes. That means the measure should concentrate in the large part, in the passing block, in smaller number, not a very large part. This set of measure has some, has the same hyperbolic rate. Yes, yes. Yeah, you put on the epsilon bar. Yes, yes. This measure is, this subset is not an input. For example, if you, yeah, this one, you can choose one epsilon bar and consider this set. And you can choose very, very freely to choose epsilon bar. This is not an input subset in general. Okay. So if we do not have this assumption of same hyperbolic rate, the answer is no. This is the upper semi-continuity is not true for C1 plus alpha, no uniform hyperbolicity. Let's analyze one example from the knowledge and the new house. Take an open subset and in the CR diffuses. And such that F8 has a hyperbolic basic site with the same adopted neighborhood U which has persistent homo-connected tendency. Persistent means we make only small neighborhood we find such map. And then take a hyperbolic parallel orbit with this one. This means the roughly say it is the reactant opponent of the point P. And then we may assume that P is both R shrinking meaning this property. And now recently recently meaning this property. From this CR small perturbation, there is an interval I of the tendency between WU and WS. Near this interval, we make one more small perturbation which we denoted by G to create a curve in this W part with n-bombs as in the following picture. In this picture, we can see that there is a following picture. In this picture, the interval from AI to A2 is denoted by I and J is this cosine function. And amplitude A equals this one and omega equals this one. This A omega R very small means the CR perturbation is small. And then we will get an n-house with topological entropy. We choose K in such a way. A times this one almost larger than one. And this is almost less than A. The meaning is this, the meaning is this. Why we take the two condition? The two condition ensures that we have this picture. That means this cosine function is not very high, not very low. It has a intersection with I in this way, in this way. And also the amplitude is suitable. And also this K ensures the following. We can for each n choose n large enough so that the topological entropy of the n-house is greater than this number, greater than this number. From a variational principle, there exists a gothic measure such that the merit theoretical entropy is larger than this one. Well, since this map explains unstable direction about n times and contract the stable direction about one over n times. So the reactant components we are close to this number which are almost equal to this one, this one. Moreover, since by attrition of G the data spends most of the time near P. So million is close to the periodic measure delta P and let n large if necessary such that this is true. This means what measure we choose is close to mu P and the components of this measure larger than this. Now we did not, we did not this one. Okay, to conclude for any D fuel G from a open side mu which is a subset of mu and the continuation of G by small perturbation we can get a D fuel G n satisfies the following property we named as n. The first property is there exists a hyperbolic basic side and a gothic measure on that such that the entropy is larger than eight. The second is the measure very close to mu P and the reactant component is larger than this. Then we denote mu n as the subset of new satisfy the property as n and denote this subset. This mu n is clearly an open property and from the above discussion it is density. So r is a denoted side. Now for G from the residue side r and the continuation P there exists a sequence of a gothic measures mu n such that mu n close to approach mu P and the reactant components greater than this. So from definition this means mu n and mu P and the mu P are supported on the same passing side with these three parameters. But in this case we say the entropy is positive but of course the entropy of mu P is zero. So in this case we find a counter example of upper semi-continuous. Well, let's see more. Not that although for each mu n is supported on uniformly hyperbolic basic side and the angle between E S and E U in this basic side is uniformly bounded by about this quantity. But if you ask n goes to infinity n goes to infinity then this angle between the stable and unstable bundle could go to zero. So in this case the two bundles have no domination. So from this example we say maybe we need domination to prove the upper semi-continuous. Now this is the definition of domination. And then we proved the following theorem. Let F be a C1 d-fuel of our compact remanifold and let lambda be a passing side with a domination splitting. Then the entropy map is the upper semi-continuous. Okay, upper semi-continuous continuity of entropy map holds for C1 plus domination but not for C1 plus alpha. Goes a little against a common intuition that conclusion are usually parallel between C1 plus domination and the C1 plus alpha. We give, we mentioned two example. One example is stable manifold theorem which is proved by Bayesian, Bernoulli-Grawizia and also passing formula, passing formula. Proved by Tian and me. Okay, let's make some remarks. The dominated splitting can be extended to its closure but the closure lambda bar may support no hyperbolic measures. An application of theorem in three dimensional case is a class of C1 robustly trans-stable partially hyperbolic d-fuel on T3, introduced by Manet. Where the Leibniz opponent of center bundle is now zero. For high dimensional case, Bernoulli and Vienna constructed C1 open side of d-fuel with dominated splitting but in this case, they atomize no hyperbolic bundles. Where the closure is supported, where the closure is the whole T2, T4 and which supported SRB measure by Ali. Well, the topological pressure, we can take this one as a definition. The upper semi-continuity implies there exists the equilibrium state, the meaning is this. Provided there exists a sequence in a closed subset of this side such that this limit, this equality is true. And this condition is satisfied by many systems with some hyperbolicity including Manet's example and Bernoulli's Vienna's example, where in their examples, the closure of a certain open neighborhood of the unique maximal entropy measure is contained in this subset by Liang, Liao and Tian. Of course, now people know much about this two example. For example, the entropy, Boozy's work and Manet's other work. Well, now we talk about the proof of the theorem. This is the definition of entropy. And then let's say we fix one measure and a real and one partition. And then this part is smaller than zero by definition and this part is continuous because the border has no measure. And this is upper semi-continuous at mu. So this is a less than smaller number. So let's say to show the upper semi-continuous we should estimate the difference of this. But the difference is less than 12 plus zero plus this difference. So to show the upper semi-continuous at mu, we need to show that this difference is small for partition, for all the measures in this subset. This is the key point we need to show. To do so, we use the local entropy. Local entropy, we know Boole's work Newhouse and someone else. Our definition is a little bit different from that work. A little bit different. So one way is we define it pointwise. We define two such entropy. One is with star, one is not with star. Okay, of course this one is smaller than this. We are positioned by using our definition then the difference between one entropy and the entropy with a partition we are less than or less than well controlled from upper side by this interval. This will help us a much because in this case, in this case, we, the hyperbolicity assumption of measures guarantee some kind of uniform hyperbolicity for site with large measure for all nearby measures. So from this, we say this one could be in a large measure could be small, could be small. And the small measure part, we can control it. So then applying domination the police limer, we can get a small local entropy. And from this one, we can get our conclusion. Well, we denote, we denote the ball. This is centered mu and the diameter is rho. And then we will talk some detail because we consider the passing side, the measures supported on the passing side with some hyperbolic property. So this means, this means nearby measures, nearby measures have this hyperbolicity property in a large site, in a large site. So a single chain of domination undermines fake fallation by brands and the weakens on the passing side. The fake fallation has two properties. One is almost a tendency. Another is a local invariance. And this number R zero is unified. So by using these local invariant fake fallations with uniform size, we can construct local product structure. This local product structure together with hyperbolic time police limer use the following. This one is the Bowen ball centered at X and with radians R with the infinite length. Well, this claim says for fixed one measure mu and sigma, there exists R rho and measure side with big measure, with big measure for only measure for many measure. This T is uniform for all the measures, for all the measures. In this case, if you take the center at T at this subset with the big measure, then the Bowen ball content exactly the center itself. So of course, the entropy with star is zero. But what we use is not the star. What we use is a little bit bigger than this star. So from this claim together with some estimation, one can prove that this one is bigger than the entropy star with star. But this one still small, still small. So let's say you see in the big subset with big measure, in the big subset with big measure the difference is small. And in the compliment which is with small measure, the local entropy is bounded. So from this, we say the difference is small for all the measures nearby T and the unapartition cosine. And so the difference is small and this proved our theorem.