 Thank you to Stefano and thanks also to to Marcelo and Jean-Christophe for organizing this school and later this conference. So I came here several times it's always a huge pleasure. I followed many courses I learned a lot so now it's I hope this one could be fruitful for you so it will be with Raphael here. We wrote some notes that will be on the web page. There are some mistakes we are correcting so if you find another one you should tell us. Okay so let's start so the subject is partial hyperbolicity so but let's start with something a bit more general so we are interested by different shable dynamics so we are on a manifold which is compact on the relays and we are interested by discrete dynamics so a different morphism and the topic of this lecture is to discuss some particular class of this different morphism because there are few things we can say about general morphism but to start with let's say something general which is how to so today this morning will be more motivations so how to decompose the dynamics the composition of the dynamics so there are several ways I will I like one I will present now which is quite natural so to decompose the dynamics what you look for our attracting region trapping region so a trapping region is an open set which is whose closure is mapped into itself the manifold u is below and it's mapped into itself and now if you iterate the region any orbit here in you is attracted by your set a limit set the attractor of you which is compact and invariant and you can do the same in the past you define kind of repeller which is also compact and invariant and the other orbit have to to go through this middle region you have you decompose the manifold in three parts a and this region o even by the iterate of you minus f of you so points here are not recurrent at all so they they come from our and end at a so it's highly non-recurrent so you're mainly interested by orbits in our or in a so that's what you have if you have one such region but then you can change the region and again you're interested by the corresponding are and a and you trash the other point so if you trash all the points in the non-recurrent part for all the the region you what we mainly is what we called the chain recurrent set so this is the intersection of m minus o for all trapping region you you is open and invariant so this set is compact in invariant so we want to understand the dynamics in this set and he said naturally splits into a collection of invariant compact set so because you you may define a relation so if you take XY in our chain recurrent set you say X is equivalent to Y if each time you have a trapping region this cannot separate X from Y so either X and Y belong to you or X and Y belong to its complement so it's an equivalence relation and the equivalence classes are called the chain recurrent classes and they are compact in invariant because this relation is closed an invariant so this is something we would like to understand in general how does it split maybe there is a only one class maybe there are several ones maybe infinitely many one it's the first first property would like to understand okay there is another way to define there's objects which explain the the word chain so chain is another name for pseudo orbit so if you fix epsilon you say that a sequence in M is an epsilon pseudo orbit so for any n the distance between the image of Xn and Xn plus one is less than epsilon it's like an orbit but you allow some error at each iteration and then r of f is the set of points that are recurrent for pseudo orbit so for any epsilon there is an epsilon pseudo orbit from X to Xn well you have to specify that you have at least two points because otherwise it's trivial and then you may define the equivalence relation in another way so X is equivalent to Y if for any epsilon there is an epsilon pseudo orbit from X to Y and another one from Y to X again so something we could discuss in the tutorial is why this two notions are equivalent so I don't know for the tutorial we will see we can I propose something we can discuss any other thing maybe one indication is if you have if you fix a point and epsilon you may try to understand this set the set of point one that you can attain from X by epsilon pseudo orbit and again n larger equal to one so try to understand how this set looks like okay so this was something general even for homeomorphism it makes sense so now let's discuss some more particular class of systems so before talking about partial hyperbolicity let's talk about hyperbolicity uniform hyperbolicity so many of you maybe have no summer but uniform hyperbolicity I'm not sure all of you so I like to quickly give the definition and then give the main property hyperbolicity so we are still with a different morphism F and we consider K compact invariant set and we say that it is hyperbolic so for any point you may look to the tangent space at X if you want to simplify you may mention M is a torus and so okay and then so that this linear space have splitting into two linear spaces the property that so you have invariance for any X the image of yes is the space yes at the image and the same for you okay and then you have so you have contraction of ES there exists C and lambda smaller than one so just for any point and for any positive iterate so the the norm of DF at X along the space yes and any rate is smaller than C lambda n so this is contraction along ES and you have sorry the definition is symmetric in the time so now if you reverse time you have contraction along EU so this is a very classical what is less standard is how to define a hyperbolic different morphism so there is one definition I like and I'm fighting to promote this definition so with what I said before it's very natural you say F is hyperbolic any of its chain record class is hyperbolic is a hyperbolic set and there is another equivalent definition it's the fact that R of F can be decomposed as invariant compact set that are hyperbolic so I'm not taking and saying here that there are the chain record classes I'm not assuming a fine place of chain records classes for instance so to see the equivalence it could be an exercise no no no here the hyperbolic city is in restriction to the chain record set I will say give you example so what is more standard is to talk about what is called axiomay so I don't want to give the definition but for those who know more but we can discuss this if you're interested it's to show the link with a more classical definition so in fact it is equivalent to not axiomay but axiomay plus no cycle condition so some some very classical examples so one case is when the bull manifold is is a hyperbolic set a hyperbolic set so we say that in this case F is another famous example is this so linear automorphism on the two torres so this integral matrix matrix acts on R2 but preserves it too so you may take the question which is a two torres and it is a hyperbolic at the opposite you may have a very small chain record set so you may have a chain record set which is finite finite union of periodic point that are each of them is a hyperbolic set so hyperbolic periodic orbit so here it's almost what we call more smell different morphism in the definition of more smell different morphism there is another technical assumption I don't precise here so a very simple example on the circle you had some sink and some sources then you may have more complicated example maybe more complicated so you have the horseshoe is defined so on the sphere you have or in the plane you have a rectangle R which is mapped onto itself this way but inside R the horizontal and the vertical are preserved so you see the the vertical is stretched so it will give the unstable direction the the vertical and the horizontal is contracted so it's a stable direction and if you take the consider the orbit which don't exit you find a contour set and at last there are attractors so you may have periodic point that attracts there are things but you may have a more complicated attractors one example is a clicky attractor I don't want to give the the construction just to say that so again in the sphere or in the plane so you have a hyperbolic set so that so it attracts any point in a neighborhood but the set itself is locally homeomorphic to a product so to contour set times the line and the dynamics so stretch along this this line so you is here and contract to understand it's not so easy to to see how to build that we did some picture in the nose okay so let's summarize some important property because it's it will be our starting point for partial light of the city so one important property is the shadowing so it says that if you continue up so the orbit so an orbit with error it is shadowed by a true orbit so for any epsilon there is delta such that so let's consider K hyperbolic set and so each time you have an epsilon so the orbit in K then there is a point in M such that the orbit of x stays close to the absolute orbit so at any rate it is delta close so this which seems maybe technical it has many consequences so an important consequence for us is a spectral decomposition because it will be one of the main subject in this in this lecture so spectral decomposition tells that if f is hyperbolic then the number of chain recurrence classes is finite finite and then so each chain recurrence class has some recurrence but recurrence by pseudo orbits in fact in the hyperbolic setting it's better so any class is transitive so not only we can travel inside the class by pseudo orbit but we can travel by a single orbit so there is it contains a dense half orbit forward orbit which is the same as containing a dense backwater so two other consequences of this lemma one is so the stability f is hyperbolic then any G which is C1 close is also hyperbolic so hyperbolic city is an open property but not only that so there is so the dynamics on their chain reconcept are conjugate there is a homomorphism which conjugate is equal to G so the dynamics remain the same after perturbation and the last property related to the shadowing is the existence of coding or we have Markov partition so there is a symbolic representation of the system f restricted to the part that we want to understand just as an example for the horse shoe then it is conjugated so to a symbolic system which is a space of sequences of zero and one where the shift so we understand very well the dynamics so then some other properties that are important so one is to understand the geometry of hyperbolic sets it's a famous stable manifold serum it says that if K is hyperbolic and if you take a point in K then you may look to points that have the same feature as X what we want to call the stable set points whose future orbit gets closer and closer to the orbit of X in the future so this is so usually it's WS of X it is a so manifold is a an image so many emails because they accumulate itself then so another property is a description of a statistical description of the dynamic so let's take f hyperbolic deformer fission in general it's difficult to understand the future of any orbit but we are happy sometimes if we understand the orbit for almost every point so this is a case and we have some finiteness of the possibility so there are finitely many probability measures such that for almost every point so what does it mean almost every point it means if you take any volume it's almost every point for this measure and used by the volume if you change the volume you have the same notion of almost every point then the future of X distributes according to one of this measure there is I such that you look to the statistic of this orbit you put dirac masses along the forward a piece of the forward orbit this is a I normalize this is a probability measure and it converges to one of this measure for usual convergence of measures and then you may study the ergodic properties of this measure I won't talk about this now and the last property is the fact that okay we would like maybe to understand all the hyperbolic system and to classify them we are far from being able to do that but there are some reasons for example so the hyperbolicity could constrain the dynamics and the manifold so for instance there is one result by you have France and you have well I didn't give any name but so let's assume that F is another and M connected and that the unstable direction is one dimensional and then M has to be a torus and the dynamics is conjugated to a linear automorphism very similar to the one I gave at some moments so it's quite free another thing we understand so in dimension one hyperbolic morphism are exactly the those I describe in the second in dimension two hyperbolic morphism so according to the spectral decomposition breaks into finitely many classes and each of these classes has to so looks like one of the example I gave so either it's another or you have isolated periodic orbit or you have a contour set like the horseshoe or you have an attractor repelor which is similar to a click in but in higher dimension we we don't understand okay do you have question so let's go to partial hyperbolicity so one goal is to relax the notion of uniform hyperbolicity to address more a larger class of system so that the way this one so you first define what is a partially hyperbolic set so again you have fixed the morphism F you take a compact invariant set and you say it is partially hyperbolic if so for any point x the tangent space can be decomposed into linear subspaces so I didn't precise in the hyperbolic setting you had two spaces the first is called stable the last is unstable here you allow a third one called the center stable and you require so invariance as before here sigma would be s you are see you require that yes and you are contracted so yes is contracted by the f u by the f minus one as before and you have a center part which has which is dominated and domination domination means I really so this is a presentation but introduction but this afternoon I will describe in more details this notion so domination means each time you take vector so for any point x you want to compare what happens along the stable so you take a vector yes in ec and in you that are non-zero and you want to understand how they are contracted or expanded and so for instance if you look to the middle part this is a reason I took it non-zero it's less expanded that along you because you have this factor and it's more expanded that along yes similar so the hyperbolic case it just a case the center bundle is trivial we don't want that to have only a center bundle but in this case there is no information but sometimes we allow one of the extreme bundles to disappear because for instance in the hyperbolic case you may have only the stable if you have a sink attracting paradigm so so at least one one of this bundle yes you is non-trivial so for the moment I won't give more example so you have already hyperbolic example what okay a very simple example a nanosurf system on M and you and you do the product with the identity on M2 and another many products and you can check it is partially hyperbolic the center bundle is the space a tangent space to M2 so one goal is to understand how to extend the previous property to this more general setting this larger class of systems I want to discuss all all the properties I have introduced but there are also other motivation for studying partial hyperbolicity it's not just trying to generalize it's also because sometimes partial hyperbolicity appears a natural class so as I say one property we will discuss a lot is a spectral decomposition so what I mean is essentially the finiteness of the number of classes but there are system there are large class of system that are not partially hyperbolic but that have on infinitely many classes a famous example is a new house phenomenon so let me recall what it is so if you start with a C2 surface deformation which exhibits homoclinic tangency homoclinic tangency mean that there is a periodic orbit hyperbolic periodic orbit oh so just so it's hyperbolic set so you have stable and stable manifold so and there is a point P in O so that the stable manifold of P and the unstable manifold of P have a non-transverse intersection point okay something like that if you think to a fixed point saddle meaning you have one-dimensional unstable one-dimensional stable the so you have two invariant manifold the stable and looking to the past you have the unstable manifold and you may require that there's a manifold that are have a tangency in this case you may check that f is not hyperbolic and what new house has proved is that then there is an open set of system C2 so f is in the boundary of you and in you so energy in a dense subset of you not only it's dense but it's a g delta dense has infinitely many things or sources so since our periodic point that are attracting sources hyperbolic periodic point that are repelling so there are particular case of chain record classes so in this case you have infinitely many classes so this seems a wild behavior not only you have infinitely many classes but also to describe the so this classes here are simple but you have many other one which are much more complicated that could appear so to describe the system is very difficult subject so you would like to characterize system which have only finitely many classes so this is something we we don't know in full generality but let me just give a statement so one case is a case you have only a single class so let's say f so what one example you have a single class is when you have a dense orbit f is transitive anyway say f is robustly transitive any g deformorphism C1 close is transitive has a dense orbit and there is a theorem bonatti lia's puja's they have something a bit more general but instead that if f so if f is robustly transitive let's assume m connected then f is not partially hyperbolic but volume hyperbolic which is quite related so it but globally the whole manifold is a volume hyperbolic set meaning so to summarize you you have a splitting like for partial hyperbolicity but here it's not uniformly contracted maybe and here not uniformly expanded but you still have invariance and domination but what you have I want detail much that you have you don't have uniform contraction here but you have contraction of the volume along this space in the future in along ECU in the past and so for instance in particular in dimension 3 your partially hyperbolic m is partially this is one case another example where partial hyperbolicity appears so you see that homo clinic tangency may generate wide behavior so high complexity so you may want to characterize a system which don't exhibit homo clinic tangency homo clinic tangency so let me state one serum so I proved with Enrique Pujols Martin Sambarino that way young so it says that in this so if you fix the manifold in a space of C1 diffeomorphism you find two regions a closed region and open they are these joints and the union is dense with a property that if you take F in T then F is limit of diffeomorphism exhibiting tangency if you take F in O then F is partially hyperbolic in a stronger sense that what I have set up to know so any any class is either hyperbolic or has a splitting so ES EU and the center part splits into one-dimensional bundles and here yes and EU are non-trivial so this is a and so this forbid the existence of tangency close so this is a good setting to try to understand all this properties try to generalize all this property of hyperbolic system I have few minutes more yeah so what to say so one goal is to extend the as I said the previous property to this larger class of system another motivation for partial hyperbolicity is that we have many example natural example that appear in various setting so Raphael will talk about that tomorrow this we will discuss on on Thursday we will talk about robust transitivity and on Friday about finiteness for attractors then what about the other properties we talked about so invariant many faults so this will be addressed on Wednesday well I hope and there are many other subject with partial hyperbolicity we won't talk at all about so other subject so there are many ones but some in one that are been much developed in the past year so the problem of physical measure so there's measure that I described in the republic setting and in particular if the system is preserved the volume you have a natural measure and you'd like to know if it is ergodic so ergodicity when a volume is preserved and another subject that has been developed is a prime of classification and maybe we'll talk a bit about that tomorrow so mainly in dimensions but if you look to the literature you will see that partial hyperbolicity appear at many places because it's a very natural setting to extend some properties and usually the first case one looks at is a case of the center is dimension one and then a second case would be to address the case you have a sum of several one-dimensional center okay so and this afternoon it's more to discuss the definition and the basic properties so thank you