 Jerome welcome everybody good morning good afternoon good evening it's you know we're all getting used to these really really international events and it's really nice to know that there are people from California to China really the following so welcome to everyone to the second day of our last week of this event dedicated to Mark with Partition and Young Towers and we are very pleased I'm very pleased to present the first speaker of today Jerome Boussif from CNRS Orsay is it still called Orsay Jerome everybody's so confused about the names of the universities well let's keep it for the over the comments since it's been quite long okay okay and he's going to talk about dynamics of smooth surface differmorphism thank you okay thank you thank you Stefano for organizing this conference and the school and the whole activity and inviting us so yeah so I will I will talk about this joint work with Sylva and Omri actually it will be a kind of running joke since they will talk about what follows tomorrow and the day after that and probably I don't know about tomorrow but certainly the day on Thursday you will understand why it's part it's relevant for the topic of the workshop okay so but yeah it will be it will be okay so we so okay that's so what the talk will be organized like this first I will recall some definitions and basic results about exponents and entropy then I will state the the main results after I will try to explain a little bit what goes into the proof and I will give an application to another proof of a theorem of your game okay let's start so the the first object you want to consider is the top level of exponent so we'll have a closed manifold with some remaining structure a smooth differmorphism on it usually you should think that r is infinite and then we'll consider the the space of environment of probability measures or invariant probability measure or algorithmic and invariant probability measures and we always use the weak start apology as as usual okay so the the top level of exponent at one point is the the gross rate is this gross rate of the iterates of the norm of the differential of the differential in some order so when you iterate forward if you have a measure then the top level of exponent of the measure is the average of the point-wise quantity of course since this point-wise quantity is invariant if you have an ergodic measure then it's almost everywhere constant and that's the value you want to think about okay so this is the point-wise representation of course there is the it's very well known for a long time that you can also compute the this average top level of exponent in the with this with these formulas okay and the the point of this is that you see immediately that the since it is an infimum of so these are integrals of continuous functions so they are by definition continuous the value of these integrals is by definition a continuous function of the measure in the weak start apology therefore since you have an infimum of continuous function you get another person in continuous function and in fact here we don't need an c infinity a c1 is just possible okay so that's the we have a person in continuity okay it's well known that you do not have a lower some continuity one way to get one standard way to get a kind of example is to consider a generic digital morphism in a new house domain so with robust hetero clinic tangency in some in some hyperbolic set I don't want to go more into details well it's well known that you can approximate let's say any periodic orbit hyperbolic orbit by ones which have weaker and weaker exponent so you you can get it's not it's not it's a person because the weapon of exponent the top level of exponent is a person continuous but not lower semi continuous okay now the other guy is the entropy so let me recall a few things so you have the notion of dynamical balls introduced by dina borg and boren and it was introduced by first to compute the topological entropy okay that's what you can call one of boren's formula for entropy and catwalk as it is well known introduced this show that you could also use this kind of idea to compute the the entropy the calligraphy entropy of an ergodic environment measure okay so what do you do you fix a scale epsilon that you will let that you will let vanish and for given scale you consider how many of these dynamical balls you know to you need to cover a set of given measure let's say one half it doesn't matter because here is the ergodic and uh you take the gross rate and you are in business okay if the measure is not ergodic you have to take the average of this quantity it's not like in the so that's one of the drawback of this thing but okay so you have this this way of thinking about entropy that you can keep in mind and the the first thing so here we also have a personal continuity it's a result of new house based on yamding theory that tells you that if you are thinking this most then uh actually the this entropy a map on the set of environmental measure is a personal continuous okay so it's a little bit like the the exponent but the assumption they are not the same and the technical is completely different and uh again you don't have lower semi-continuity here it's even easier as soon as you can approximate a measure with positive entropy by let's say a periodic orbit then you see that it is not lower semi-continuous okay so we have these two players the entropy and the exponent they are both upper semi-continuous but not lower semi-continuous so and now the of course it is known that they are really classic relationship between the two I mean in particular real algorithmic quality okay that I have recalled here that tells you that uh entropy is bounded above by the exponent uh well I should add a a further plus here this is the maximum of the top level of exponent and zero and okay the today what we are going to see is that for c infinity so for c infinity surface determinism actually we have the a similar relation but between the continuity of the two quantities so what do I mean by that okay so let's start with an ergodic case of our theorem where we can see it's more easy to understand what's going on so you have your c infinity surface decomorphism you take a sequence of ergodic invariability measure okay by extracting if you like you can assume that it converges quickly to some measure new okay now you you make the assumption which is not automatic that uh you this limit is ergodic and what we claim is the following that okay you consider the ratio between the limit of the entropies of the new case and the entropy at the limit of the limiting measure okay so of course this limit doesn't have to exist so you but you can uh further extract to to make these two limits converge and now there is a real assumption okay which is that we we want this limit we need this limit to be positive okay so it's a it is something that will come regularly and that is really necessary for what we do uh that we need the this entropy not to go to zero okay so this is what we can assume and now we have these two ratio since the functions are both I mean the entropy and the top level of exponent are both uh upper semi-continuous these are numbers between zero and one okay and we claim so that the continuity of the entropy is not worse than the continuity of the Lyapunov exponent okay what do I want to so this inequality in some sense I mean if you formulate the problem in that way of course it is uh sharp in the sense that okay if you give me two numbers alpha and beta which satisfy this inequality so okay less than one also okay you can find uh uh examples which satisfy so that you you observe these two ratios okay so that's the that's the the main result in the algorithm case okay let's see some consequence so the first consequence is uh like advertised as advertised that if the along the sequence you're considering the entropy is continuous in this sense simply that so the of course it means in the previous theorem that the ratio will be one and therefore the second ratio for the exponent also has to be one so this is what we call entropy continuity of course for because of the examples I gave or I suggested then well in most case you never have you never have true continuity of the exponent but we have continuity in this sense which we call entropy continuity that is if you have a sequence of algorithm which converge so that the entropy converge then uh then the Lyapunov exponent has to converge too okay so why this will be important in the for the rest of of our work that silver and then we will present it's because it is a kind of rigidity in high entropy by which I mean the following that okay still in always in the setting of infinity surface the homomorphism let's take it topologically transitive this positive topological entropy we know by our previous result that there is a unique measure maximizing the entropy it is so then if you take a sequence of measure uh every big measure which approximate the topological entropy okay so first by a personal continuity uh you see that the measure in uk have to weakly converge to this unique measure of maximal entropy and then when you apply the main result you see that not only the you have weak star convergence that is convergence of all the of all the averages of continuous function but you also get convergence of the Lyapunov exponents okay that's the another nice uh uh simple application is uh to Hausdorff dimension so if you have a measure you can well I don't know why I wrote it like this I think it should be invariant measure for us but okay it doesn't matter for the definition the Hausdorff dimension of the measure is the infimum of the Hausdorff dimension of the sets with full measures with full measure for that measure okay so you have a function on the space of measures and now what we say is that this if you restrict to the set of ergodic measure with entropy bounded away from zero then the Hausdorff dimension is a personal continuous okay the the proof is uh to use Young's formula we are in dimension two probably this Hausdorff dimension and then use our result about uh well continuity our main result yes sorry can I ask a question sure sure if anybody is welcome to ask a question no I'm just trying to to um you know to to digest this your entropy continuities so maybe maybe you you mentioned something in the previous slide so if your sequence of ergodic measures is um are all Dirac deltas on periodic points then uh the conclusion holds also right so it means that I know because you're assuming that you're well you don't know it's it could be well but but this fact that the limit has is positive entropy is part of the assumption of the code yes yes yes yes we we we need that it's an assumption uh so having the entropy uh the limit of the entropy to be positive by raising equality is enough to to get the positivity of the of the limit of the exponents but the assumption that the entropy is bounded away from zero is really an assumption okay because you see you could imagine that in fact you for instance in the problem of household dimension that you would get uh uh that you would have measures with bigger and bigger dimension but but it's because the ratio between the entropy and the Lyapunov exponent goes to one but both are going to zero and this we cannot say anything about and you could imagine that okay is this ergodicity assumption really uh necessary well we can pretend to do away with the ergodicity assumption of these measures I'm not speaking about this one for now uh but here you see that the assumption would become that we want almost every ergodic component to have positive entropy okay and since you go to me I will confess everything at once uh you see this set is not is not necessarily well you there is probably it's not a compact set because here I'm assuming ergodicity and I could probably weaken that to uh I could remove ergodicity but then I would have to say that all the all the ergodic components uh have entropy bounded away from zero and then I will trouble again okay so so so so so so let's say it's mainly for advertisement purposes but but so in your in your in your first corollary the entropy continuity yes uh okay so now now this is for for different morphism right so it doesn't apply for example to causatic family and so on you don't get anything like this right so I'm thinking well I'm not sure we don't we do not claim it I think it's possible but we do not claim it yeah as you said is it is what we say is for infinity surface so but so suppose you have a sequence of periodical bits uh converging to the srb measure for example yes then then then this corollary holds right then you're saying that the no no oh no because this yes because the the the entropy of the srb measure is positive no but you need the entropy of the measures that are converging to be also positive and even lower bounded oh you need the entropy of each new k also to be positive yeah yeah you need the limit yeah not not just the limit to be positive well it's not the entropy of the limits so it has to be because we get the assumption is that the I call but I see I see I see so it's part of the assumption is that the e that the limit of the entropies is positive which means you cannot have all zero entropies in here yes it's it's uh when you want to think with seriously about srb measure that's uh that's a big issue but so if you're if you're a pro solid if I continue I don't want to take up too much I tell you the time but if you're approximating by horseshoes if you're approximating the srb measure by horseshoes then this will by you know by by by uh uniformly hyperbolic invariant sets then this would apply this corollary right yes I think that yeah if you have fatter and fatter horseshoes uh then uh where with the entropy lower bounded so they are really fatter I mean they're well the entropy should be lower bounded this is what I can say and yes it will work right okay okay thank you you're welcome so yeah by the way uh for everybody I didn't say but uh questions are welcome uh yeah so now we let's uh do away with the ergodicity assumption on the limit which is not very natural uh okay so now we still have a sequence of ergodic measures uh that converge okay this is so sense for free since you can always extract but the limit is just some invariant measure uh and then you have a statement that I did not want to show you at first because it's a little bit uh for me at least it's a little bit difficult to see whether it means something or not uh so you have a decomposition of the limiting measure okay this is what is written here so new zero and new one are two invariant probability measure you have uh some beta which is just the mass of new one and then what we claim is that the limit of the top exponent uh of the new k is beta times the top exponent of new one okay for this same number of beta and the limit of the entropy is bounded by beta times the entropy of that same new one okay so if new is ergodic then of course uh ergodic measures being external points uh these measures have to be equal okay so new one is equal to new so you see here the first equation just defines beta and then uh if you well the assumption is still there positivity assumption is still here so you can divide and you get back the the result of the of the it is really a generalization okay we see that it's so why one uh one a nice application of this generalization is uh let's say this contribution to the thermodynamical formalism of srb measure so always you have to fix a threshold with this approach on the entropy but then you have equivalence between existence of arsenide, elbow and measure uh with entropy at least h and the fact that if you compute this kind of uh well i don't know if i dare to call it anthropological pressure so here you see this difference is the measure theoretic pressure of an invariant probability okay with respect to the geometric potential okay and what we do here is that we take the supremum over all the ergodic invariant measure uh with this bound this lower bound of the entropy and what we say is so what is known uh this is piezine's formula is that if you have an srb measure then uh for this measure you will have that this quantity so the pressure of the measure is zero okay which is just to say that you have uh equality in real inequality so that's piezine entropy formula so this this is this uh let's say that this this is piezine okay or some version of piezine if you want to say piezine was talking about volume but it's piezine formula the what we prove is the other direction just so then you have to i don't want to then it's an exercise to see that okay if you will be able to get uh you will consider a limit of a good sequence of measure uh and then you will see that you have to see that the sum of the ergodic component uh will have to to satisfy the the piezine formula and therefore the srb measure okay so that's uh let's see sorry why is this a corollary of the theorem how does the srb measure come into the theorem oh it comes because here you see that you get so yeah okay i forgot a little bit about it so the you get a subsequence that converges yes certainly this converges to zero then you you decompose the converging measure right yeah and you get one and you get an integral of the pressures okay and you know that this integral is zero uh from the this inequality okay and what you and then you you look at the at the at this ergodic decomposition you know that's okay you have measures which are not hyperbolic okay but they cannot be uh sinks because uh where you are taking a weak starting limit of measure with positive entropy okay so what remains is measure that would have a zero exponent and a positive exponent or two zero exponent okay but this one they have so they they can contribute uh what am i saying uh okay sorry sorry anyway you you look at this ergodic decomposition and you check that uh you have a part which is where you just see zero equals zero so this is this is this doesn't give you an srb measure that you don't care and the remaining part has to compare in other srb measures okay so sorry i'm a little bit too excited now but yeah so that's so now let's uh unless there are other questions or comments about the statement now i would like to explain the proof so the the proof use some ingredients which are quite uh standard like this trick of using a projective extension of the dynamics okay which goes back to the to very early study of the exponent okay i want some names and this is very this is the basic of okay so what what is it this projective extension is just above each point x we consider the set of uh one one dimensional and linear subspace so the set of directions okay and and then you get so you get another smooth manifold of dimension three and on which you have the the extension the projective extension of the map is this f hat which is what you think okay and now the what you can define is the duration of the of the map of the differential uh in a not only on a point but uh in a given direction just with this formula okay and since we are in dimension two and we will be uh looking at hyperbolic measures of set of type we are really looking at one dimensional stable one dimensional unstable so uh the the expansion uh when we look we are going to look at into is one dimensional okay and therefore it's just an additive process in the term of this uh function five which has the the very uh nice property uh of use but nice uh of being continuous on this uh on this projective extension okay so now when we when you're looking at an ergodic and hyperbolic measure okay by hyperbolic I always will mean uh uh set of type then uh you can say okay do I have lifts yes I have lifts that's this compactness but I can be very precise about the lifts so in fact you have two invariant uh measurable graph so the unstable the graph of the unstable directions and the graph of the stable directions they are both defined over some some measurable subset which has full measure for any hyperbolic measure uh it is just measurable okay that's uh usually that's life that's actually why we have something not to say and then you have each of these invariant graph gives you uh so which is defined almost everywhere in particular with respect to you defines uh unstable lift this new hat plus and a stable lift in this new hat minus okay just uh you just leave to the to the to the one of the two graphs okay the graphs are invariant everything is fine okay and now the connection with the Lyapunov exponent is that the top Lyapunov exponent is the average of this of the dilation uh with respect to the lift the unstable lift okay and of course you have the same type of formula for the bottom Lyapunov exponent which you can I mean I leave the definition to your imagination okay it would just be the average of the same function to the with respect to the unstable lift okay so now we have these these measures and this continuous function and the Lyapunov exponent are uh averages uh of uh a continuous function with the right lift at least in the ergodic place so now when we we have our little problem with a sequence of uh measure ergodic measure on u k converging to some measure on u so we leave the new we want to deal about with the top Lyapunov exponent so we consider the unstable lift okay there is some k that should be here uh okay and perhaps after extracting uh it will converge to some measure on u hat k which will be an invariant probability for the projective extension of f of course by continuity of the projection uh between the this projective extension and the initial manifold u hat is a lift of u okay so what we can what what does it so for now we are in some sense just a nice translation okay what is what does it say say it says that uh the limit what is the limit of the top exponent of the new case okay each exponent is just this average okay since now it is an average of a continuous function it is by definition of the mixed out topology above uh the average of the limit function okay so now there is no more discontinuity for what is going on okay what is going on is that we didn't say which lift is new hat so everything is is encoded uh here okay so in the ergodic case we can be it's easier to to to see what's going on when new is ergodic when the limited measure is ergodic then by this theorem or this this classical lemma uh we have we know that we have only two lifts two ergodic lifts sorry of new okay any lift is a combination of the ergodic lifts so it is okay this is what is written here it is a combination uh with some also alpha convex combination of most so alpha is some number between zero and one okay what does then when we write this what do we see that the limit of the exponent okay we do some the trivial computation which i hope is still correct okay and so you you have uh okay you have the you and you should compute new hat of phi so this is one minus alpha new plus hat of phi which is lambda plus of new plus alpha times lambda minus of new okay then you here it is we are looking at several type things so lambda minus of new is minus it's absolutely values and you you do your thing the way you want to to to present thing here to get close to the formulation of the theorem i make this coefficient beta appear and this coefficient beta is this is this strange number okay so what is the the takeaway from this little computation it is that so this quantity here or the fact that beta is not equal to one the defects between one and beta the difference between one and beta okay what what what accounts for it it's the fact that we had measured carried by the unstable graph is gamma plus and in the limit some of the mass alpha in these notations leaked to the uh somewhere else okay but the only place it can go uh is the well in fact this is not they are two two places either it goes to contracting or it can go to to to anywhere when the two exponents are zero okay so that's the that's what we what that's the interpretation of this of this thing and now now we want to use that so now what is the the the game the game is to use the discontinuity of the exponent which we understand as this leaking of mass you can also think about it as some kind of a quasi-homoclinic tendencies to say that okay we'll have this leaking of the mass we want to from the stable to unstable okay so what does it mean concretely it means that you are looking at the unstable and there are points there are some measure where it gets very close to the stable direction okay it corresponds to the fact that we are looking at a non-uniformly hyperbolic system so the unstable direction okay most of the time it expands but sometime it contracts okay and this and here we are looking at a limit of ergodic measure so it means well if we see in the ergodic decomposition of the limit a negative exponent it means that there are very long stretches of time where new k-generic points actually or new hat plus k-generic points above they they represent a direction which is expanded most of the time but for very very long time they they get very close to the stable direction to a stable direction and now it contracts for this very long stretch of time and now it goes to to some other part of the measure of the space where it it expands and and at the end we have this combination of the two exponents okay so now the the game is to use this to get above on the entropy of the measures new k for k larger okay that's what we want to try to do any question yes so I'm just trying to see how do you deal with the fact that mu might not be ergodic with respect to the assumptions you stated before oh so here what I will just say is that uh well uh actually we won't let let me so what what I'm going to tell so actually it won't appear explicitly that's a little bit of the I don't know the magic of stupid algebra or something like this it will appear in this beta so let let us so it's so for me it is helpful to to think about the ergodic case because then I understand what is going on I understand exactly where things come from in the general case it's it's much more uh it's more complicated so we don't we won't need to to make it completely explicit what you could have in mind is that okay you are converging to an ergodic to a measure with a natural ergodic decomposition this ergodic decomposition think about it as being finite okay it's measurable so okay you can break it up okay what does it mean then to converge to have an ergodic uh measure very close to a non-ergodic one it means that for very very long time because the ergodic because the limit the ergodic component of the limit are invariant of course so they correspond so when you get close to them you could also form relate this in terms of sets uh open set approximating uh in measurable invariance of set because of the invariance of the limit when you get close to it you you stay there a very long time but still if the limit is not ergodic it means that after some time you escape and you get close to another part okay so in the ergodic case when down there it is ergodic what you what you expect to see is that for a very long time you are close to some compact subset of big measure uh where the graph the expanding graph is uh continuous and you expand at a definite rate and if you look at the right iterate given to you by i don't know the depth theorem or just the definition i gave up to top gap enough exponent you will see expansion at that rate and then you have very very long intervals where you are close to the unstable guy to the stable guy sorry and then you see contraction at the rate uh given by the negative gap enough exponent so now you you have many many ergodic components so you you you will see a combination of all these things so i'm trying to see if the decentralization i'm trying to develop is true you explain to me now why and if the limit is not ergodic you can find geometrically sets which carry interesting ergodic components but i wonder if you consider the measures it's just a couple of the topical number exponent and entropy no no no well what happens is that well generally if you look at the at the lift of an arbitrary invariant probability measure what you have is just that uh the over each ergodic component you have a lift the for the ergodic component which are hyperbolic you just have the combination of the stable and unstable lift okay for the other component it is more complicated it depends on the cases you have there are there are works on classification of these things where in sometimes you you you have no decomposition sometimes you still have some kind of parabolic phenomena so you have only one direction to which is uh invariant so it's complicated we don't want that we focus on the hyperbolic situations but you then you have a combination okay and you don't necessarily have all the stable or all the unstable if you have a mix between the two so it gets complicated but it will turn out that we don't need to get too much into this into into explicit what is going on okay thank you but yeah but that's the the idea is this you are converging to an ergodic decomposition this ergodic decomposition is invariant so you see the various uh behaviors with uh this the proportion given by the decomposition but you see them uh not i mean you see each of them for a very long time yes thank you you're welcome so okay so that's the weapon for uh to not to have to deal with this problem or at least to in a in a rather simple way it is to uh okay so we consider pieces of orbits and we say that they are neutral if you do not see any expansion or if you see only very small expansion so gamma is a is a small uh positive number so i don't know it is much smaller than something okay early is bigger okay the main thing is is gamma so what we what so remember phi is the dilation here what i have written i don't know why there is a this is here is to say that okay when you iterate uh one time two times three times any any number of time until the from the beginning of the interval until anywhere in the interval what you see is a contraction so it's like a hyperbolic time okay but it's hyperbolic not in the sense that it gives you that it gives you uh expansion but uh in the sense on the contrary that you do not see expansion gamma is small okay so in the ergodic case what will happen is that you have these stretches where you contract because you have to you are very close to the to the stable direction and uh i should make a little picture okay so this is this is time okay this is it's okay this is a very very big time okay and now there are intervals where you see uh contracting behavior okay let's call it lambda minus okay i have a i have a pen it will it will be easier for everybody okay you have this lambda minus okay and then so okay you need to consider some iterate and when you but once you do that essentially you see all the time you see a contraction by this lambda minus okay so therefore you see a contraction when you start when you continue okay here you only have contraction so i'm only interested in what's but now when you when you go further you you have some time where even if you're in lambda plus okay let's say that all over here you're in lambda plus still when you uh start to to do your your your business from the beginning here you see globally a contraction okay this is exactly okay this corresponds to the fact that the liapunov exponent the negative liapunov exponent will cancel apart of the positive liapunov exponent okay this is what we saw in the little algebraic computation where we had this this factor of something like this okay we had not only the mass but okay and when we have something like this okay we call it a natural block okay so for technical reason we don't want a strong contraction we tolerate a little positive contraction and we certainly need to have a bigger intervals the one which and this should not be a problem because we are converging to something invariant blah blah blah so now what we what we what we introduce is uh okay so when we have this think about the generic points for u k for the major the aquatic measure on u k okay and we what we will cut from what we will select in them are these natural uh blocks okay so the natural blocks they have a nice property that if you have a union if two natural blocks intersect then the union is a natural block okay just because here you have one so everything is good there okay but if you if you have another natural block from here okay then you you you say okay until this point i use the fact that the first interval is a natural block and now i point so so the the average between here and here is uh is small and now when i want to go further i use the fact that the second interval is is also a natural block to complete to see that between this time and any of these times i still have this uh uh no expansion this neutrality okay so so therefore you can always take the maximum if you if you wish uh in your mind to have a clearer picture you say okay i can always assume i have i'm looking at a maximum natural block okay i'm considering measures for which uh the the exponent this is the top exponent it is not a sink that i'm looking at so therefore the average for a very long time will be positive so the i cannot be in an infinite natural block i can just take the the maximum natural block uh that i see and they will be finite okay and so i can define what it means for a point to be in a natural block or segment and this is this sky alpha n okay and now i can uh i can make i can frighten you with a formula uh which is in fact not so complicated which is home yes what is the alpha quantifier sorry what is the outside alpha is gamma okay okay i got used to use in the paper we use uh alpha but here alpha was used for this click so sorry yeah yeah thank you thank you okay so now okay i see the tiny is running out i don't want to so now what we see is that so what what we do is we have this uh typical uh c is generic this is generic point for new k hat plus okay this is this x hat we we look at the we fix alpha sorry gamma l uh so that we have an ocean of natural blocks we see the maximum natural blocks and we only keep those okay so it's like an empirical measure but you only keep those things okay and you you find a little bit uh okay here might be another mistake okay so let me uh paper over it okay so you you you select these natural blocks and you take some limit okay and you want the the the parameters to go so that you get get uh you get uh longer natural blocks okay and uh that the the expansion you tolerate is weaker and okay so let's say let's pretend that okay this formula is not completely correct i try to so in this way you get a measure at the end okay if i didn't put this i should get just get uh the measure on new k hat plus here i will get some as which converge to the new hat to the measure new hat so so here i would get new hat if i didn't put that okay here i will get a measure which is part of new hat which is uh so i've dominated by new hat okay and uh yeah uh which okay and then now what i so what is the natural decomposition uh is to say that i can do that and uh of course so okay i need some condition which will be more or less automatic if i'm looking at generic point so since time is is is uh evaporating i will not talk about this and now you okay you have of course you need to go to a subsequence but you can ensure that you have a natural limit okay this uh sub probability this invariant sub probability m of zero hat that i tried to define before and of course you have new hat okay so which is which will be this yeah okay so the first thing is to see that you can make these things converge okay it's an exercise in uh in a diagonal argument uh and things like that and now you you just get your decomposition but above where it makes the most sense just by saying okay new zero hat so one minus beta is the mass total mass of this natural part okay what remains will be beta times new one hat okay of course if beta equals zero or one you have some some things that uh are not well defined but you don't really care and now what are the properties uh the properties are that first okay the measure uh the natural limit it may it is made of very long so first you you you selected very long blocks since these blocks are very long this was the parameter l that goes to infinity the measure you get are invariant in the limit when l goes to infinity okay so you get invariant probability measure this is what i should have said at first so the first thing is that you you took you have when you have a neutral block the average of course of your dilation is less than gamma which is going to zero so it is less than zero but you took these intervals to be maximal so in fact it's not less than zero it is equal to zero and you can check that it goes to the limit okay the condition is that if beta is equal to one new zero hat is not really defined okay so you don't don't pay attention to that really and the second thing is that you can exhaust uh all the zero thing and you will get that for the other guy the new one hat almost everywhere uh these uh limits are positive or if you wish the ergodic component of new one hat have a positive defined uh something which is expanding they have a positive exponent okay so you have this decomposition okay that's the that's the where the decomposition comes from okay and now i have two minutes okay so this uh let's be so now we are going to apply this to our entropy estimate so we do like in catholic theorem but along unstable manifold okay using disintegration along unstable uh variation okay that's we are going to use so the smoothness will be used to uh to work uh following yondin by using parametrized curve and not that analytical ball especially we will use curves that uh with speed that never vanish that never vanishes so we can lift them to the uh projective bundle okay and the uh so now we have this uh this piece of curves inside of which we want to keep track of a set of positive measure uh for this disintegration that there will be a set of measure which are typical for of points which are typical for the measure okay so now what is the idea you you apply your dynamics okay and you see how things separate and essentially now the goal is to use uh the neutral blocks to say that during the neutral blocks uh nothing happens you okay this is clear in the tangent dynamics okay that's the definition of the neutral blocks you expand almost no you have almost no expansion okay so if you start with something small the diameter will not expand okay so but you have to be uh so I won't talk about yondin so the first thing is that you you will uh oh sorry I put it so no the first thing is this epsilon hat so it tells you that you want to have a small diameter above so that the the curve is almost straight uh yeah it doesn't the the tangent doesn't get uh wild okay so you will be able to control the expansion by looking of the piece of curve by looking at the derivative at the speed anywhere on this very small piece this is the epsilon hat and now that you have control below you okay this control below with control sorry this epsilon hat above will tell you that you control the expansion below uh from the dilation okay which is uh small since you are in neutral blocks but of course you you can do that once and then you will start to you you want to do it again but you have lost control maybe of what's going on above it is not because you are contracting below that uh above it doesn't start to oscillate after some time so this is where you use the second parameter epsilon to say that okay most of the time so I am always in the in the measurable graph uh the unstable measurable graph okay but most of the time I am in a compact part of it okay and therefore I uh can assume that I am uh looking at a continuous uh graph most of the time there are both there are exceptions maybe I've got the theory uh and now we use this modulus of continuity to say that okay if we take epsilon very very very very small given epsilon hat we will uh still control uh the what's going on above uh after applying uh some iterate of it okay and in this way we can see that when we are in a neutral uh block we do not have to subdivide to subdivide uh our small curve very much I mean some number but which will be exponential with a small exponential like is usual in young and young in theory is behind this capacity of going from small diameter to small number of smooth parameterization okay so now okay since I am I think I am already over time so I will uh just uh tell you very quickly so we we have built so we started from this sequence of measure we built at the composition above okay and this and using this decomposition uh and the argument I alluded to on the slide before we get something like this that is a cover of a piece uh of unstable manifold with a significant measure okay by a small number so this is where we use the fact that we do not grow during the natural uh blocks uh we see we have this beta sorry so what one minus beta is the natural blocks and then during time beta something else we have to use something else what we use is that we are copying the measure mu one so but only during a fraction of time beta so you have this uh this uh beta factor here okay uh and we still keep the the size of everything small okay so really once we have this smaller this representation by curves with small images we really have uh uh control on the on the entropy by this fiber the cathode formula okay and the the conclusion of all this uh is that the limit of the entropy is bounded by what we want okay that's half the theorem actually is the big half the second half is that okay what is the limit of the of the exponent okay but the limit of the exponent is mu hat of the dilation okay but in this uh sum the first term gives you zero okay we we saw that it's essentially the average of our natural blocks so it's a zero of a maximum natural block so it is zero so this there is only these terms that remain which gives beta times uh this long this uh top exponent of mu one okay and when we okay to do this seriously not only I need to say that mu zero hat gives a zero contribution to the dilation or the average but I also need to say that mu one hat of phi is the top level of exponent remember be careful here it's not a new plus or something like that so the reason for that is the other the last property in the natural decomposition that almost every energetic component of mu one hat was hyperbolic so because of that it tells you that uh you have uh it is really the top exponent that you are seeing all the time okay not sometimes not sometimes okay so so this gives you the the the general statement okay so I so Stefano how long do I have you can take another couple of minutes if you want but we don't want to be too late yeah okay okay so let me just say it very quickly then so recently that it did get uh proved uh a fantastic result which uh is just there on the slide and uh okay after he told us about that we realized that actually uh so he has a stronger result than what I have put here on the slide but this part of the result which is still is nice we can also uh we we also we obtain another proof by applying the repatriation arguments okay that I have tried to to give you some idea okay so so we are using his result to make you read our paper okay so yeah I don't want so it's uh a preprint on archive uh as I said uh Sylvail we will tell you uh why it is so it is related to the to the workshop and so thank you thank you very much uh okay well we we can take a couple of minutes for some questions if there are any I have one quick question uh is this um um is there any chance of doing these kinds of things for those the obvious kind of question in three dimensions so I think that when we when you are in uh co-dimension one it's possible yes that's so when the unstable the dimension is one uh probably is doable okay you you see that here everything happens because we are working on curves okay so we have something that is uh additive once we are on curve uh so it looks like it shouldn't be too much trouble I mean we hope we haven't done it and we haven't uh okay maybe there are some difficulties but it seems reasonable beyond that it looks much more difficult and thank you and I also have a question Jerome yes yes actually I have two questions one is similar to the question you know I was just asked and in your paper you you explain why your result is not true in hard dimensions if you take a surface and a product with a system with many zero entropy measures so you would have many and measures of maximum entropy but what happens if you impose on a higher dimensional system that the measure of maximum entropy is hyperbolic then I assume the the problem would not be with the correctness of the statement but how to control stable manifolds geometrically so I wonder if this can be overcome so yeah I'm not sure I'm not sure what what do I okay so so France probably you you need okay you have these examples where you have measures of maximum entropy of different indices okay like these circle boundaries uh studied for in particular by uh for the guest earths for the guest earths as a bn us uh so here you have uh measures which uh yeah so so if you just so then it you need something then so yeah but what I would say is that it's possible that uh yeah you need to this okay you need to do something more and uh also you you need to restrict to put some instead of just assuming that you so for instance in these examples you can make them work if instead of just assuming something about the entropy you say okay I want to fix the index okay then for this example which are very special and in some sense they evade your question because they are partially hyperbolic recenter dimension one you can get uh you can get uh you can get a lot of things but uh yeah I don't know in general but what is the the obstacle what prevents you from extending this result to high dimension I don't know I know the technical obstacles which are as I said it's we are doing something one-dimensional so probably it works in any dimension if you have only one uh positively a point of exponent uh it's but going so so this I'm not saying I know how to do it but it looks like the same type of techniques uh could work at least might work but then in general I don't know if you are really I really uh so yeah you you yeah I don't know if I would like to to say okay let's assume that you have a situation where the measures of high enough entropy are hyperbolic maybe with a fixed index then okay the this could be a setting where it might work but I don't I don't really know it's okay so for example if one fixes an ergodicomal clinic less like in rodriguez so the gives us a new list and the index is fixed and you have the topological transitivity that you need for uniqueness and then the question is yeah then yeah okay if it's uh yes yes it could be but I don't really know it's and my second question is even more annoying and you explained how to make the reduction from the case where the limit is not ergodic to the case where it's ergodic it was it was quite geometric and I wonder if you tried to find the more naive reduction or once you you found the reduction you stop because it feels like perhaps one can well in this in this setup it seems like you only consider measures which are identified by they're identified by the level of expanse and the entropy so you could just divide let's say two measure classes and you take some measure and divide it's ergodic components in such blocks and then if you have a sequence converges to a measure you have a subsequent which accumulates in a block and the size of the block can be arbitrarily small and you find in the end a subsequent just goes to an ergodic component or something slightly more naive like that I wonder if it's if you have a if you thought about it and it's not possible well I don't know I don't know well I just want to say that I'm a big fan of this program so thank you for this talk well thank you for your questions and your attention thank you very much um okay so thank you