 begin. Okay, so welcome everybody to the third day of the workshop, Markov Partitions and Young Towers. So the first lecture of today will be given by Janu Chen from the nice city of Succo, so he works at Succo University. And he will talk about Markov Partitions for hyperbolic systems with singularities. So please Janu. Okay, thank you Yuli. First of all, I would like to say on the organizers for giving me these opportunities to play the recent work. This is a drawing world which found one of Hong Kong Zhang. We are constructing the Markov Partitions for hyperbolic systems with singularities, but mostly those kind of systems are like video systems. So most of the talk actually I'm going to establish the video setting. This is the outline of my talk. As I said, first the setting of a uniformly hyperbolic system with singularities is kind of complicated. There are five standard assumptions extracting from paper by Janu from Hong Kong Zhang. So I would like to give you a warm up on why these assumptions are needed. So I would like to record something about the Sinai Dispersing Billions. And then I will give the general assumptions. Then I'll give a kind of a short overview on the current results which are closely related to the Markov Partitions, Yantau's and the statistical properties for BDR-like systems. And then I'll present our main results and give the ingredients of the proof. So here's where the Sinai Dispersing Billions come from. We study Laurent's ideal gas model in an ideal way. So basically we look at the molecule gas like a single particle and we'll try to study the motion of this particle. Well, when there's no collisions, it's just moving in a unit speed. But if there are some scatters which are just other gas molecules, then you may elastic collision. But to make this model a mathematical problem, actually we need to add assumptions, something like we would like to say, well, those scatters are arranged in a periodic way. Then we can actually just look at the face base, no face base, look at the table like this. So we are just looking at one single chamber which is identified with two torus, but there are some scatters inside. Like here, we have one scatters which is inside the two torus and then four scatters on the four corners. Then to study a motion of a BDR ball, so we are looking at the gas molecules like BDR ball, we actually just study the collision map of this BDR ball, meaning that we study the trajectory of collision points. So in this picture, on the left, we can actually coordinate the orbit of this BDR ball using two coordinates. One is the location on the collision place of the scatter. Another is the refresher angle with respect to the normal of the scatters. So this is kind of a standard setting. Then of course you'll say there might be some disconnected scatters, but ideally you can actually glue them together to make a circle in the base. A line for the refresher angle will range from minus pi o2 to pi o2. Now for collision map, you just send one collision point to the next one. So we actually denote it by x equal to r phi to the next point, s1. But here you see there's actually some normally effective collision like the grazing. So for instance like this, so at this point there's a grazing. This grazing collision actually creates some discontinuities. So on one side, if you go a little left, it applies on this center scatter. But if you go right, it goes away. So we actually have some singularities, preliminary singularities, SGL on the angle plus or minus pi o2. Now for the collision map, well geometrically it's hard to study, but actually using some differential geometry, it is easy to derive the difference of the temporal map. Actually this formula is actually derived in the book by Chandler from Macaulay. It's a very standard formula. So let me explain the parameters here. So here k and k1 are just the absolute curvature of the collision points on s and s1. And the tau is what we call free pass, which is the length of the string connecting s and s1 in the table. Now if you look at this formula, say you are looking at just a purely point of the temporal map, you will see that this matrix, 2 by 2 matrix is actually hyperbolic. But there's some unpleasant effect causing by this factor, which is cosine theme 1 on the denominator. So it means that when you do a near-gracing collision, the tensor map will blow up. So in our words, the tensor map is only defined outside this set, we call s1, actually here I should write this way. s1, which is just the preliminary singularity, union which is the preimage of SGL. Now the situation is that because of this tensor map blows up near s1, then the distortion control is hard to get. So here's the idea by Sinai in which he added some what we call homogeneous singularities. So let's go back to the picture. We are going to add some additional singularity lines, which are new store lines, which are given by the pi over 2 minus 1 over k square, where k is a large number. So those are what we call homogeneous singularities. Now with those homogeneous singularities, we can somehow regain the distortion control, at least in a local way. Now we are going to combine SGL, the preliminary singularities and the homogeneous singularities all together, and you know you buy this COVID SGL. Okay, so this is basically the setting of Sinai Dispersing Billions. Now you see the basic capacity comes from the distributions or say the structures of singularities. Also we need to control the tensor map near the singularities. So to generalize this Sinai Billion to some general setting, we actually consider some piecewise invertible map on two-dimensional compared to most manifold. So of course here we need to add several assumptions. Well, to be honest, when I first learned those assumptions from Professor Hongkong Zhang, well it was kind of complicated to me. It's like three or four pages of those assumptions, which is for me there is kind of difficult to understand if you just look at this thing for the first time. But still I need to go through all these assumptions so that we can move on to our main results. So first of all, let me emphasize that we only consider two-dimensional manifold. So for higher dimension, I don't see that we can generalize easily. Now also this will require that this map is piecewise invertible on piecewise. C2 is most actually you can vk to c1 plus alpha. And also because for the BDL we have a tiny washable property, which is kind of useful sometimes. Now we need to add the flowing assumptions. The first one is uniform hyperbolicity. The second one is kind of complicated but crucial, which is about the structure of singularities. And the third one is about the regularity of stable unstable manifolds, which is actually kind of standard in differential dynamical systems. And the fourth one is also, well, we assume that there's SRB measure and we take a missing component. And the last one is, I guess it's again a feature in BDL systems. It's called one-state expansion. So let me explain term by term. So first of all, to represent the uniform hyperbolicity in a BDL systems, what we can do is that we can establish the family of stable stable comms. So those comms are a continuous family. And then when the derivative or the standard map is well defined, then we should assume that those comms are invariant, meaning that the RZM comms should be forward invariant and the state comms should be backward invariant. Also, a vector in the unstable comm will be uniformly expanded by the vector Nanda. And a vector in a stable comm, well, if you take the derivative of the inverse map of t, then you'll be expanding again by the vector Nanda. And also, here we assume that the unstable and stable comms are uniformly bounded away from 0. Now for synod dispersing barriers, we can actually give a precise formula for the unstable comm and stable comm. Well, it's just, if you go back to the face space picture, it's just an account which represents all the vectors which slope greater than the curvature by finite. This is the unstable vectors in the unstable comms. And the stable vectors are just the operatives. So here I'm drawing the unstable curve and stable curves, which you see the unstable curves are the red ones which are increasing and the stable ones are decreasing. Those green ones are decreasing. So this is the uniform hyperbolicity. And the second one is a little bit complicated. It's about the fine structure of singularities. So as I said, so we take the curve S0 to be the preliminary singularity, namely pi o2 or minus pi o2 plus all the homogeneous singularities. And then we take S1 to be S0 union with t inverse S0. S minus 1 will be the other way. So what we can see here is that the S1 is exactly the singularity for the collision map. And S minus 1 will be the singularity for the inverse map. And also we need to assume some structures on those singularity set S1 and S minus 1 where they consist of finite or countably many curves, smooth curves. And also they are uniformly transverse to the stable unstable comms. And if you disregard the S0, meaning pi o2 and minus pi o2, the curves which are new singularity from S1 should be stable curves, meaning it should be decreasing in the phase space. And S minus 1 should be unstable curve, meaning it's increasing. Also the each curve in the singularity S1 or S minus 1 will be terminated either on S0 or on the singularity S1. Okay, and there's another assumption is that the collision map, although we know it blows up near the singularity S1, but it should not blow up too fast. So it's actually controllable by the distance of the point to the singularity S1 to the minus Q's power, where Q is the number less than 1. So this is the second assumption on the structure of singularities. Okay, now the thing is that we can iterate the singularities a little more times to get the singularity of tn, meaning the nth part of the collision map, which we denote by Sn. And also if we iterate backward, we get the singularity of the inverse map, the nth part of the inverse map. Now also we take the union of all four singularities like Sn and take the union from Jl to infinity, then this would be usually for BD assistance as plus infinity would be dense in the whole phase space. Now also if we take S minus infinity, also it could be dense. Now what's happening in this situation? Well, again we are only looking at a two-dimensional case. So there's a argument by Chernoff, where he actually used those singularities to build some curve linear polygons to approximate the stable mathematical manifolds. So in other words, the local maximal stable manifold would be exactly the open kinetic curve in the M, which goes as plus infinity, we move. Now the local maximal unstable manifold would be the other way. And so you see the stable unstable manifolds, local manifolds will be can be Rp short in the phase space. Also those short curve manifolds could be very dense. Okay, so that's the consequence of the structure of singularities. But I guess union was in two-dimensional case. Now let's talk about the third assumption. Well this assumption is kind of standard in differentiable dynamical systems. So now we have families of stable manifolds and unstable manifolds. We write to add some additional assumptions, but not to restrict. So first of all we write to a show where those local manifolds have boundary curvature and boundary lengths. Well this is always doable because we are actually considered compare manifold. So we can always chop long stable, long unstable to small pieces by adding some additional singularities. Now for the second one, the third one statement, well this is kind of standard. Well first of all we would like to say that along the unstable manifold we have the low Jacobian is locally holder. So by low Jacobian I mean this is the Jacobian we should respect to the Lemanian volume induced on the unstable manifold. So this Jw means the Jacobian we should respect to the Lemanian volume, leaf volume. And also we would like to say the stable polynomial is absolutely continuous. But here we need to be a little bit careful because as I said there would be some short stable unstable manifolds. So even when we take two long unstable manifolds W and W2 close to each other, well the stable polynomial map are not defined on a whole unstable manifolds. We can only, usually they are just a subset. So let's do it by W star one and W star two. But anyway, so unless we can define a stable polynomial, well this map is absolutely continuous and the Jacobian is a boundary holder. And the fourth assumption is that we would like to assume that the system has a missing SRB measure. So what is SRB measure? Well it means that the conditional measure on each unstable manifold is absolutely continuous we should respect to the Lemanian leaf volume. Now, of course in our setting we did not assume anything. So it could be that there are several SRB measures. So we can just take one of the SRB measures and take the basis to be our phase space. So this is just a technical assumption. And also we can take missing components. So why don't we just assume that this SRB measure is missing. Okay now, now for back to the C9 dispersing beer you see that the SRB measure is exactly a smooth measure which is equivalent to the volume. So this is the precise formula cosine phi drd phi where the constant c in front of it just normalizing factor to make this probability measure. Okay now the last assumption that is kind of specific for BD assistance what we call we call a one-stage dimension. So the situation is well due to the singularities maybe I should draw a picture. This is a W a local unstable manifold. Then when you apply the collision map it might be cut by the s-1 into 590 many or counter remaining pieces. Some pieces are long some pieces are short. Now we call those components v alpha. A name the pre-image of v alpha we denote it by w alpha. Now this one-stage condition is essentially saying that if you look at the Bayesian iterations it tells you there's some some of the Bayesian contractions for an unstable curve which is short enough. So here we have a parameter s over here saying that well we tell this sum well this is loose wheel but I'll explain it by C9 dispersing beers. So this is kind of like a weighty sum and this weighty sum will be less than one if we take a small unstable manifold which is small size let's say less than delta. Now as I said if we take s equal to one meaning in a situation of C9 dispersing beers well the situation is much this formula is much better to understand. So we just take the Bayesian meaning the quotient given by w alpha divided by its image v alpha. So this is just a Bayesian contraction rate for the this component and then you sum them together as long as the unstable curve is short enough then this sum will be less than one uniform less than one. That's the setting of one-stage expansion. Okay so finally I'm finishing the all the assumptions. Now besides the C9 dispersing beers what kind of Bayesian satisfy these assumptions? Well you can look at conservative perturbations of dispersing beers. So those conservation perturbations sometimes are caused by the external force or sometimes you can consider some non-elastic reflections with kiss and slips. Okay this kind of perturbation will be considered by Chernoff and also Mark Demers and Hongkong Zhang. And also when we are doing dealing with some non-uniformly hyperbolic beers like Bony Moichi beers or semi-dispersing beers sometimes we can actually take a nice or say good subset and take a punker return map to get the induce map to satisfy those assumptions. So for example if we consider a Bony Moichi stadium like this standard one then we can actually take the subset to be the left half circle and the right half circle and then you see when you induce the dynamics in these two half circles they actually give you the uniform hyperbolic systems. The induce system is uniform hyperbolic and also certify the other conditions on singularities. So the punker return map of some like Bony Moichi beers or semi-dispersing beers also certify those standard assumptions. Now okay now let me give a brief overview on the market partitions and young parts and also statistical properties for BD assistance or some sort of closely related to BD assistance. The first one will be still be for the C9 dispersing beers. In 1980 turn off so maybe I made a mistake here I think this one is Bony Moichi and C9 in 1980s they constructed a countermarked partition for plain C9 beers with some restrictions and then in 1990s Bony Moichi turn off and C9 they actually constructed the market partition for plain C9 beers still with finite horizon for a large class also they use some new method but unfortunately this countermarked partition only give you the symbol coding by countable states. So in our words you can only look at the C9 dispersing beers as a countable mark of shift but without some additional structure you do not know the decay rate at all. So they actually use some geometry analysis to obtain the stretch as part of the decalculations in 1991. Now drastically improvement is actually due to Lai San Yang in 1998 and 1999 which is nowadays called the tool for Yang Tao she actually used these kind of tools to prove the exponential decay of correlations for C9 beers with finite horizon but without common points. So as we know Yang Tao is not a market partition it's actually just a market extension. Now in the same year turn off actually extending the tools of Yang Tao to the case of infinite horizon and also he could do for the case when you have common points. Well this is for the C9 beer case. Now for the non-uniform hyperbolic beers and it's actually much harder. So after Yang published her paper in 1998 after five years well Macaulay actually made a breakthrough using Yang Tao to prove that the Mugunimoji stadium has a polynomial missing rate of this is of this order but this order is just upper bound it's not an optimal bound. And then also in 2005 turn off Hong Kong Zhang established the polynomial missing rates for semi-dispersing beers and bonimoji beers some bonimoji beers but still they can denote gallery of the logarithm term in an estimate. And then in 2008 they improved to the optimal order which is one over n. So this is the results related to closely related to the BD assistance. Now well if we look at the Yang Tao's there are actually two ingredients. So both require hard work. So one is the construction of the base meaning the hyperbolic product set. Also we need to verify all the assumptions of Yang Tao on the base. And then the other hardware is on how do we do a tail estimate meaning that when the tail is exponential when the tail is polynomial for concrete systems sometimes it's hard to estimate. So right here let me concentrate on the construction of the base. So there are two words that I want to mention. One is by Chen Lofeng Zhang in 2009 they actually posed this standard assumption h1 to h5. Then they used dogopias, synopias method to prove the coupling lemma and hence the exponential decay and central limits theorem and so on. And Yaluo is also by MacDemos and also Hong Kong Zhang and they prove actually used a different method meaning they use transfer operators on some anisotropy from her space and do a functional analysis but still get something like existence of SRB measure exponential decay and central limits theorem. So let me draw some major comment for the Yang Tao which are suitable for BDs. Actually well in Yang's paper she constructed the Yang Tao for synod dispersing BDs which finite horizon. And then Chen Lofeng Zhang also have some ideas but they do not use the language of Yang Tao but still where they still need is the base. Well the base is kind of like a hyperbolic plotter set but in Chen Lofeng usually actually like to coil a magnet because it kind of extracts a lot of unstable manifolds when the unstable under the iterates. Okay now for the construction of Yang Tao for BDs usually you see the high function it's usually just a stopping time it's not necessarily to be a first return time. Okay now I think it's a good place to present our result and I made some remarks. So back to our assumptions we consider a two-dimensional compact smooth manifold and then we have a piecewise invertible I forgot piecewise smooth also tiny reversible map satisfy those five standard assumptions. Now under these assumptions we prove that the collision map the map T means the countable marker partitions of Rpc small size we should expect to the SRB measure more work this kind this partition has a sponsor tail by sponsor tail I mean well our partition is essentially Yang Tao but the return map return time is exactly the first return so it's the first return Yang Tao and then we look at the return the tail of return time is exponential okay so our improvement is on the situation is that we actually construct the map partition instead of the Yang Tao which is more efficient. Now I think it's a good place to make some remarks and actually I would like to ask some questions for Yuri and Carlos. So Lima and Matthews actually they constructed countable marker partitions for a uniform or non-uniform hyperbolic BDMS for general hyperbolic measures meaning high hyperbolic measures so they use the technique symbolic coding technique originally developed by Omisari so but I'm not sure so Yuri I'm not sure what's the situation right now do you get some tail estimates for the this kind of partition or no and we also are not able to get anything about the existence of measures of maximum entropy so this is actually a difficulty because the measure could be concentrated towards the singular set okay I see I see so Baladi and Demers they have a work that shows the existence and uniqueness but for that they use the machinery of anisotropic spaces so it makes it much more complicated and long nevertheless they are able to prove the existence and uniqueness. I see I see so yeah I have another question because in Matthews talk actually this is the first time I know you actually weaken the assumptions by Chen of N'zhan to some Chen of Essians but I'm wondering what's the difference between those standard assumptions in Bilius and Chen of Essians I believe our assumption is kind of stronger but I'm not sure if the Chen of Essians also implies the assumptions H1 to H5. I think it does so actually what Chen of did was thinking about the Bilius in order to develop his axioms okay the difference is that in Matthews talk he only required a uniformly C1 plus Lipschitz regularity of the invariant many folks and the original Chen of is just like as you stated requires bounded curvature so also Chen of axioms they work in any dimension and they're quite general so instead of for example the one step expansion that you require he just requires the growth lemma to be satisfied of course the question is when is the growth lemma satisfied okay I see I see I see ah so you can apply this essence to any dimension yeah his work is multi-dimensional okay okay isn't this Balint and co-authors that extended Balint and co-authors also check the conditions of Chen of okay I see I see okay it's good to know because this is the first time I learned this kind Chen of Essians which can could be much weaker than lose a lot much weaker than our assumptions which is much which is good yeah okay let's move on so now I'm going to present a schedule of proof so can I ask a question so hi so you said before so I guess two or two questions one is that I guess you get the mark of partition because your young tower is the first term yes yes yes okay and the fact that you have exponential tail means you have exponential decay of correlations is that right yes yes all right so how does that match up with the previous results you talked about some polynomial decay of correlational was that for the boom evolve which uh yeah yeah so the thing is that for boom which stadiums or the general boom which uh billions you can first find uh goods goods are set such in that the first return pattern first return map is exactly satisfy our assumptions so you actually can show our market position for the first return map and you leave to the whole space you still get the mark of partition for these uh no uniform hyperbole billions but in this case you do not have exponential tails and no no exponential tails the tail is the mess is still the same as before by Mark Julian or Chen Nofen Zhang the the tail bones are stays the same the tail yeah so which one of your assumptions h1 to h5 is not satisfied in the boonimovich billions uh not satisfied at the uniform hyperbole city right of course of course yes of course okay okay okay thanks okay yeah okay so uh the proof of many doubts is actually contains four states the first day is actually uh already done by Chen of also like some young so it's just a construction of a hyperbole product set or say a minute and then the second state is that we would like to decompose those uh this many or hyperbole product set into s assets union of s assets uh there there's quite a lot of way uh here we are using coupling lemma and uh then we are going to construct what we call a perfect minute which is like kind of similar to the nice uh property of uh the talk by uh professor pacing on Monday and then finally we see those perfect property then we can actually prove the first return is exactly a mark of in our words the young towel that built over this perfect minute our star is uh first return young towel so first uh this is kind of well known by last year also channel you can look at the channel of my client's book so we will use it for bd assistance or some general hyperbole uh systems with singularities we can construct hyperbole product set uh hyperbole product set which are just formed by a family of uh relative long stables and relative long unstable so usually our notation will be uh gamma u represent the family of unstable and inter sandwich the family of stables now also because we already have a reference measure which is technically srb measure mule so we require that this uh hyperbole product set of positive measure now uh the reason that we can construct such a set in the bd assistant is actually due to the so-called c nice fundamental lemma so it's actually flowing so if you take a point x which is not in a s infinity not in a singularity so uh for any q which is between gelon one for any a constant a we can actually get a neighborhood of x unstable neighborhood of x such that the for any piece of unstable leaf well there should be long enough stables close this uh unstable leaf so in ours there are a lot of points which were denoted by y such that the size of stable so rsy denotes the size of stable stay out of this unstable manifold which is greater than uh a multiple of the length of w and those points will off uh problem measure of portion greater than one minus q so uh in ours if you shrink the neighborhood size of neighborhood you can actually get more and more points on this unstable which long enough stable manifolds so in this way you can also require that the density which is better as a measure of this magnet of this hyperbolic product set is uh close to one so for instance like close a greater than 0.99 excuse me genuine yes so this statement holds for every a for every a yes uh because it looks like you are saying that everything almost everything has a fairly long size yes yes so uh well you see it only it also depends on the size of neighborhood so if you tell very very short unstable lane there are a lot of stables which are uh relative long but as long i mean they can be like a thousand or a million uh yes lane in this case the this neighborhood will be very very small okay because w is a small one okay yeah i see yes thank you no problem okay then the second step is that we write to use the uh technique uh of standard pairs and standard family uh developed by the gopia to do a coupling so that we can define the in this way we can define the assets of the a magnet or a hyperbolic product set so but here when i need to recall some definitions of standard pairs and standard families uh so roughly speaking a standard pair is simply uh unstable manifold carry a probability measure such in that this probability measure is absolutely continuous we should respect to the uh here we use the conditional measure of mule on this unstable main for w and also we require the locked uh locked reason of the laden dignity derivative is uh c gamma hold well c and gamma adjust some face constant and we'll begin a length uh what's this is standard called standard pair uh meaning just unstable manifold taken uh some uh regular density probability measure now uh the standard family simply uh combinations of uh standard pairs in a way that we have well maybe uncountable uh many of uh standard pairs then we need to index it by some index say this eight and then we need to uh have some measure we call lambda on this index set such that we can uh combine these uh standard uh those standard pairs together to get a new finite measure and this is what you call standard family now uh for our coupling lemma we also introduce a so-called generalized standard family well in this part we do need to fix uh hyperbolic product set up and then to we take a standard family restricted on this hyperbolic product set and take its unsupply image this kind of this object is what we call generalized standard family well you can see is that it's still uh a combinations of several pre-image curves length we've taken some uh measures on each curve but those measures are no longer regular meaning uh they are not standard pairs for each curve they don't take standard pairs but unsupply images of standard pairs okay now uh well usually when we do coupling actually we take two problem standard families and then couple them together but here for our case we like to construct multiple petitions in this step we like to study the the asset so how do I get the assets well we do a self coupling so we consider the family of the unstable family of the same the family of unstable manifolds of the magnet itself so let me do a call hi it's just gamma u interstellar gamma s so we take the unstable family as our underlying curves for the standard family and then we just take the sub-measure mu meaning on each curve we just take the conditional measure of mu to do a standard pair then we have a self coupling lemma saying that we can actually decompose uh this standard family into a sum of the generalized standard families so here w and mu are generalized standard families meaning that they are just the image of some standard family interstellar with the magnet and also uh by our by our assumptions we can also make sure the image of uh mu meaning the generalized standard families of index n of index m their image under the tm push 4 on tm push 4 they have disjoint supports if n is not equal to m and also uh the coupling lemma actually also guarantee us we have uh uh exponential tail meaning that the measure that we do not couple after n steps will be exponential small okay now we should do several coupling lemma we can simply define the support of mu n to be the asset because you'll see it's just the image unsupply image of some the unstable of the unstable family gamma u taken with some measure now the image of tms is exactly the support of tm push forward mu n so and of course here uh there might be several components of uh the n step as a subset so we can actually go on to decompose it into several components by those components i mean if you go on to compose them then the the italy of collision map on the solid rectangle containing each of them will be smooth uh here i forgot j here is the c2 smooth is tendable to the solid rectangle and similarly we have the we can also decompose the user sets into several components and then we have some a destroying uh the composition of sr set as well as user set and they are actually cross bound moving to each other so that's the second part second step and the third step is that we would like to uh get a new hyperbolic product set such that it satisfies what we call perfect property so what's perfect actually this is mentioned by uh professor pacing in monday's talk well he has a nice domain uh which have uh periodic points on the diagonals corners so uh in that case you see the four iterates of the stable boundary will never enter the interior of the rectangle and the bevel bound uh bevel iterates of the unstable boundary will never enter the interior and for this property we will call it perfect but here we do not uh actually require the diagonal corner points to be periodic points okay then this is what we describe a solid rectangle to be a perfect rectangle now if the if we have a hyperbolic product set then we can actually take the solid rectangle carry this hyperbolic product set if this solid rectangle u art is perfect then we call the magnet also this hyperbolic product says perfect now uh the thing is that we would like to construct such a solid rectangle first and then we can uh get to a perfect uh magnet then we are going to do a coupling number again on this perfect magnet together as i said use us in that case uh once you have a perfect magnet those as i said and use us and we actually give you the firstly term decomposition so uh let me just give the idea of how to construct such uh perfect solid rectangles so first of all uh when we have this uh when we get a magnet we have a poly product set we actually already get sr sets now we can take two sr sets which are compatible so maybe i should show you the picture so here we take two sr sets one is labeled by a liana is labeled by b and for you i mean the solid rectangle containing these sr sets and then to also by symmetry they are also uh corresponding uh to the a there's u sr sets and b uh use us for a and b now the situation is that we would like to get uh can consider a purely point corresponding to a symbolic holds which is like this now you see it means that uh uh this is a purely point stays in the cylinder a for quite a lot of times there's a large time m and then it goes to b then go back to n so it's a it's a point like this it stays in the cylinder the a part for a while then goes to b and then goes to goes to b and then goes to a immediately now the scene is now once we have uh this kind of uh purely orbit uh one n is large enough then actually we can prove that for the so we can prove that once we have some point of like this purely orbit go back to this square also in the rectangle meaning the intersection of u s and u a u and then uh if we have point go back here like the unstable manifolds and stable manifolds or of this point would be long enough to cross both uh in the u direction and s direction and then we can actually take a smaller a smaller rectangle formed by sjl an f sjl which is the f is the return map for the original with uh magnet we look at this uh left rectangle and then we can count there may be several uh points from the orbit of sjl uh go go inside this but it's fine because they have non-stable non-stables which you can cross these smaller uh rectangles so we can draw these uh rectangles to get uh for example because there are only finally purely points there are only finally many intersections so for example i can do this part and this part will be a perfect solid rectangle i can simply take this small piece which is a solid uh perfect solid rectangle and then we can take this solid rectangle and combine all its non-stable and non-stables to get new magnet which we denote by r star and this way this uh magnet r star is actually perfect now of course we have point mu measure because it actually contains the original magnet intersection this solid rectangle u star okay so that's the third step now the last step should be easy to understand now if we have a perfect uh rectangle a perfect magnet then we do a couple m again then we get the decomposition of s star set and u star set then you can see by this uh boundary meaning the perfect condition also the boundary restrictions well the return uh mark of return has to be first return because if it is not if you look at each picture if you say you have s m state you actually intersect the the rectangle also intersect the magnet but not fully crossed then there should be the boundary a piece of the boundary this s boundary will go inside the interior of the perfect solid rectangle s m state so this is the first step which well it's actually easy to understand but you have to consider several cases this is just one of the picture okay okay that's all thank you very much so let's thank Janyu for for his talk and I believe we have time for some quick questions so let me just ask a quick question here so do you have expectation to do something similar for the budimovich stadium uh well depending on what kind of question so here we are technically only uh useful for like srb measure because you see we are using coupling but for the coupling technique actually the limit of those standard families are just srb measure so uh i'm not sure how to do it for a general hyperbolic measures using this technique but even for srb for srb you can get a kind of mark of partition for the induced brief yes yes for the srb first we get the sr we get mark of partition for the induced map and lane because the induced map in our situation for budimovich stadiums is just the punk line first returned so we can just just leave you again it's still a mark of partition okay thanks but as i said it seems that we did not get any new statistical properties in this way because you already taken care of by my son yeah i'm also turned off for the polynomial decay way yeah yes just one question so what what um why exactly where how would you use the assumption that you already have an srb measure you should be able to replace that assumption by whatever you know properties you're using of this srb measure so the thing is that actually those assumptions are like srb measure assumptions simplify assumptions if you uh made some assumptions on the derivative or second order derivative near the singular singular set then by the result can talk and stretching actually you can construct some srb measures if i remember correctly so we if you have some more controls on the derivatives or second order derivatives near boundary we can actually control the srb measures right and i guess i guess you probably would need some assumptions on the recurrence to the singular points right yes yes yeah so those are implicit in the existence of the srb measure yes yes we can we also need recurrence yes right so okay yeah yeah okay okay so we are a little behind the schedule so let's uh thank jan you again thank you okay i'll just allow okay and just to allow people to have some water and to switch between speakers let's start in five minutes okay