 Thank you, thank you, so I want to thank the scientific committee for for an invitation and So what we're gonna work upon is some problem I kind of learned recently which is So you have some dynamically defined set invariant set like some some hyperbolic attractor for example or some limit sets of some discrete group and What you want to understand is that the hardware dimension of this invariant set And specifically we want to give some dynamical description So the thing is This is pretty understood In the conformal case meaning that the derivative of your dynamics is a homotasy in some sense But when when your dynamics is non-conformal this is This is pretty open until maybe very recently I guess like a few months ago For some cases So what we're gonna work on is We're gonna pick some discrete group so matrices So here I'm gonna pick R or C so the C case actually provides some interesting examples and So this is a discrete group and we're gonna look at its action on the projecting space say Or other grass mania and spaces like key K planes Right so Under kind of mild conditions on gamma There's gonna be There's gonna be some space especially invariant set Which we're gonna call the limit set Minimal closed and gamma invariant So I guess I should put this this here at least first actually so that this makes sense So as I said with my conditions, there's only one of the sets this statement of Benoit But okay, we went about you so so we have this set anyway, and then Okay, one of the standards exactly is this has to be mentioned This this guy for So this has a Romanian metric, right some metric invariant under the compact group the maximum compact group of these guys like the angle for example And so what you want to give a understand this is the dynamically understand this number So maybe write it down as some critical exponent or whatever this means So Let me introduce some notation and so basically this can kind of explain just Why unless the equal to this is the non-conformer case, right? So Let me use this to use some notation. So Let's pick some elements and Let's pick some norm on KD. So This inner product It's gonna be the usual inner product if K equals R Some admission product for K equal C So you have this you have the singular values, right? These are real numbers I'm going to order them So what are the singular values? So the first one is the norm of the operator and The product of I of them is the norm of the exterior power of the operator So another way of saying that is that you pick your ellipse you pick your ball of radius Your sphere of radius one You're gonna push it by your Your element and this is gonna be an ellipse and these numbers are the axis the length of the axis of the ellipse so for example Say for example that It is what we call maybe proximal say that you have a gap maybe say like this So if for example, right if you have a gap in the first index say so You can consider these spaces, right? You can consider the the top axis of this ellipse. So this is gonna be what it is Let's go right in this way of this ellipse, right? So we will do it like this so you have your ball and So these are real numbers you have your ball and this goes to some ellipse This has some axis So this is sigma one Sigma two three and sigma three So this is gonna be you one and what's gonna be a let me just you find You did minus one of the inverse, right? It's gonna be the pretty much of this This other axis of the space spine by the other axis. So it's gonna it's gonna lie here, right? So there is an an autonomous a go into some more autonomous set One of them is the one that maximizes the norm and the other is this I'm gonna call this space here So so what happens here, right? So you pick a small cone around this Autonomous direction to this space. So this guy is going to them Disattracting guide you one of she right? but it's gonna be a So I'm gonna be a ball, right? It's gonna be some sort of ellipse depend around this guy So it's gonna be something like this Right and the axis so So what happens when I look at the action? in projective space So what happens I look at the action in projective space? So this is some Some hyperplane right and so this cone is some we can think about it as the complementary of a neighborhood of this hyperplane The project device hyperplane right like that, right? And so when I apply G Right this cone here when a project device is gonna be some some ellipse, right? Of course So it's gonna be kind of near This point Which is your one of she and the axis of the ellipse are gonna be ratios of the topic and value and the other ones, right? So this set is gonna be causing you pretty this way This this is how you you contract in sympathetic space, right? So what you say say that if you're your element lies in in some district group as we want to understand, right? So the limit set is gonna come More or less over there if she's very big right probably you don't know basically what you have to understand is so How this intersection of the ellipse? Understand this intersection of these ellipse with this limit set right and kind of the position of the ellipse occupies in this This limit set and understanding what is the relation of the intersection, right? This diameter of intersection Will relate to through the axis of the Ellipse This is a kind of situation, right? So there is some contract to our picture. I don't know if anyone actually did this construction or not But it's some number called the affinity exponent so there is some Some sort of broken did it like serious and Which depends on all these numbers and And the critical exponent of this broken did it's a series is expected to be this household dimension of this set Unless the situation is very degenerate But I'm gonna talk about that. I'm gonna kind of start telling you the examples that we Will minus to say something about so let me Let me start with something that is known which is Specifically the conformal case in this case, right? So D equal to right? So for example PSR to see is the the asymmetric group of h3 up to some finite covering And I'm gonna pick some group here and I'm gonna make some definition, right? So this this guy is going to be convex co-compact. It's a definition that I'm gonna make so there are several ways of Defining this which are all equivalent. This one is This embedding so this function here this embedding function It's a question symmetric embedding So what is this? this is saying that distances of the images of this guy are Comparable of course he by lipches or just comparable say to intrinsic distances in the group as an abstract group another way of saying this is like You can look at this quotient and the show this inflow here It's No wonder he said it's compact. This is another way of saying this And the last one Is what I'm gonna do is when I'm going to consider the limit set in the index in the boundary So there's several ways to define this but it picks some point I Look at its orbit and I look at the closure of the orbit When I add the boundary at infinity Right so this boundary at infinity in this case is just p1 of c or r in this case we can do it for our truth and Why come back to come back because the convex hole So the complex hole of the set is Gamma invariant is a gamma invariant So this is always true of course and it's action and the action is The question is come basically what you're doing is you look at the limit set You feel the wish of the six so this is your gamma invariant set and You're requiring that the action on this on this invariant set because the question is compact So so what do we know in this case is a term of Suleyman Right once a Suleyman say so that the house of dimension of this limit set is the critical exponent of the group or because they did like Is it a political entropy of this flow for example, but we're more interested in this critical exponent approach So this number can be completed recently So if you never see this, let me just say a word which is so this series is conversion for big S So you want to look at the smallest S that makes this serious conversion and this is this point And you can write it down like this also right so this Exponential growth rate So this is typically conform a situation right the action the boundary is conformal so it sends a ball to a ball so Here what you have to understand in this case is Which is not easy, but Suleyman did it thanks critical exponent Ah the formula distance from all to Gamal the distance from all to Gamal Yes, so there What you're doing is We pick some ball of radius t Inside h3 and you look at how many points you have of the orbital phone this ball And you look how this the exponential growth rate of this of these numbers Pick some point Exponential doesn't depend on a point So we're gonna take kind of this example and Put it in some non-conformal situation So maybe I don't want to raise that drawing So what we're gonna do we're gonna The first thing we're gonna do is to put our SL2 in some SLD So this is the first thing we're gonna do Ah, so it's this So no the thing is I'm gonna tell you so So this this series so this is a function of S When this is very big this conversion And you look at the smallest s that make this conversion. What no this is the theorem Okay, so the idea is that so you see you have so okay, so here that maybe I should say this in this specific case So the thing is that this distance So you have your own gamma or you look at this point at infinity, right? So this is H3 And so this is going to be some sort of your you one You have somewhere else The repelling guy Right, so what you're gonna do is so there are two parts, right? Has no dimension. It's kind of easy to bounded above and hard to bounded below So what actually is going on is that these numbers? This So as I was saying, so say you pick a small ball here and you look at the complement and you push it by gamma, right? So this is going to give you here some small neighborhood of this guy And what is going to be the the radius or diameter of this guy? It's going to be something like that Which I said it's actually it's actually what I said right there. It's in the conformal case and so So in some sense you find the covering Right with it with the right Diameters what you have to understand is as I said before the the position of the limit set actually The the position of the limit set with respect to this to this ball, right? priority the limit set could be Very bad with respect to this ball. So it's called this Kind of this complex co-compact co-compact condition is kind of saying that when gamma is very big This guy kind of goes through the center That's more or less what's going on So this would be the upper bound the rubber bound is harder and you have to introduce some measures some patterns on Sullivan measures So in some sense, yeah So in a very In a very broad sense, maybe so what we realize is there's been some non-conformal situations This proof kind of goes through and this situation for this whole is an open condition So this is what's interesting in this case. I think so let me let me let me give you some example So let me let me tell you something that we did So we're gonna put this SL2 here and in SLD. So there are several ways to do this So I'm going to pick this to be irreducible So what do I mean by that? So the action, right? So let's call it style this way So this action This is equivalent to say that this action Has no invariant subspaces So there is one action like that and there's only one So this is some lee lee algebra stuff. So we can do it by hand actually So like you can consider KD You can look at it as a span so you can look at this Homogeneous polynomials in two variables, for example say that the span of the minus one So say say that the one is zero one and the two is one zero and so your action of some SL2 guy in KD, you're gonna look at it in this basis as How do you act on this basis? You're gonna act on each of the of the variables So this is just some definition But so the reason is so the point is there isn't a reducible action of SL2 in SLD There's only one So it's kind of the interesting way of putting SL2 in SLD So the thing is so here an element was the action on perceptive space was conformal But now when you look at it in SLD is no longer conformal It's gonna be some ratios some some powers of the singular values that you had before So what I'm so what do we do? We're gonna pick some compact for compact SL2 action of some gamma in SL2. So this is complex co-compact I'm gonna put this irreducibly as I said before Right so what we did is to show that when you deform this there is also some sort of formula like this that's going on So this is a statement maybe say So this is a theorem say So there exists a neighborhood of that So this is an open set in this space right in the space of morphisms open such that So what's gonna happen here? So we're gonna look at the action so let's pick some Roger there for example when you look at the action of Ro gamma in some grass mania say So this is gonna have some some limits there. So this denoted by L gamma K So this is so I got my one is a is a limit set in the positive space and I got my case a limit set in a K place So what we what have you proved? We put the this half of dimension Of this guy It's the critical exponent of some series So what is the series now? It's gonna be the ratios of singular values with the with the right guy So this is gonna be so another way to say this is a system to be the limit stable infinity 1 over t The number of elements in the group such that The log Yes, yes, yes for anything. Okay, so this looks like a perturbation result But in some sense it's not So there are kind of three steps and the proof So one step is that there is some condition that we call for an a so just say some any any gamma say any representation like this So I'm gonna write this word though. I haven't defined it so then So let's say two for me then the half of dimension of The limit set in the projective space is the critical exponent of two over one So I haven't defined this. This is important. I Will say later. So so this is like Understanding surely once proof basically So the second step would be like the two for an a is open and The third step to actually come to this theorem would be like for every K The representation lambda K row So there is issue so So this you hear from this right so there is the neighborhood of that guy such that For every K so this can be used in this kind of different ways one is To say that Disequalities and they're going to say that this equal So for example For example, so this this is as a corollary We obtain some some result that we proved with with Rafael In I think 2014 maybe 15 so Something that we proved with with Rafael is that when you take a serve will take a surface So this is close surface of genomes greater than or equal to 2 Close connected oriented as you want so when you take this closed surface and you you put it in SLD R Right And here what I'm going to do is I'm going to say so this is the connected components belongs to the connected of Of that guy so here the formation is arbitrary large So here I'm going so what I'm going to do is I'm going to put it like this But now I'm going to permit myself to go as far away as I want right this is what's called a hitching representation What we proved with Rafael in this case is that when you look at the critical exponent of Crazy Let's say that this is hk of We prove that we look at the critical exponent H1 of Ro this is constant a what you want something that we prove with Rafael Trust also 14 so like You take a closed surface you put the nsl 2 then you put it in SLD and then you deform as much as you want and Then you look at the ratios of the first singular value and so this is a very case Look at the ratios of the case singular value and the K-1 singular value This is just this is constant number that there is one So in some sense So we was something completely different or not and so it happens. So this is a very typical. This is a very Stray situation at least for me because all these limits says Well, not all of them, but it's consequence of what for a considerable he showed is that this limit says are really kind of regular They are c1 for example, for example, such in the French the French show is that for this kind of examples close surface and are So this limit set L1 is a c1 curve. Yeah, because so so I said this is not a perturbative theorem This is this is this theorem. Yeah, so I should say this I was I was discussing I was discussing with with a name Moncler and he told me that Moncler and Olivier and Toulouse and they have a similar result of this similar proof of this fact using the scouts of the mention approach. I Don't I don't know what they do. So another consequence is that the scouts of the mention is is regular It's analytic actually on this small neighbor. Oh, yes, sorry. Sorry. So let me give you So let me give you some hints of this proof in some case some special case So I think so let me so a month ago. I thought I thought I thought I thought we had a theorem, but Now it's not a term anymore So now it's a question if you want So now if you want is a question we have some arguments. I think it's true, but I don't know so question So what you're gonna do is okay, you're gonna You're gonna pick this guy, right? You're gonna start with something like this And you're gonna push it as far away as you can, right? As long as you always remain far from them having cohesion is on eigenvalues and I think that this holds for that situation consider Representation that can be connected to the standard one To this standard representation that can be connected to that guy. Let's call it principal just to say a word So we see the greater component, but I want to ask more I'm going to ask that the path So the component of what of this principal representations But the path is always Through representations that don't have cohesiveness on eigenvalues. That's not Again value coincidences. Yeah, so here I gotta put a surface here I gotta put something come back to something up here. I gotta put a surface then You so this is then this is you This is my question So here I know that this so this condition this this condition that I wrote obviously holds Well, not obviously, but it holds in you So I'm gonna push this condition as far away as I can I know my what I think is that if you if you started with a surface close surface Then in all this big set, this is going to hold in particular More specifically, I think that this is going to hold This is going to hold, but I don't know So specifically means that the limit cone doesn't touch the walls so Principal is this This is definition just to give a name because I'm using it all the time over here. No So, okay, so here The labor is telling me saying that The whole component is like like that, but it's a very deep theorem And here I am over C here. I want to do it over C Compact but this is over C this over C So what I'm going to do is I'm going to pick some Representation like this, but now I'm going to do it over C Let me put it over C Because this is this is what the actual question is and the S is closed surface and then I'm going to deform it as much as I can with One restriction, which is In all the paths that I make There's no wagon value coincidences for any element and the question is This does this imply that formula and I think it does A month ago, I would say of course now. I don't say now I say I don't know the thing is I don't think there are deformations If the manifold is closed, I don't think there are maybe someone else No, if you do this, so if your grandma was was was was co-compact here right So it's over C say close to probably three manifold Okay, you can do this, but I'm not sure that I mean this I'm not sure that you can actually deform it inside here I'm not sure Something to something have to show for surfaces or for for this open guys. It's much easy. It's much easier to do it You can't do it by kind of my hand the manifold is closed not locally. I don't know Yeah, so this is the thing right in this kind of situations, right? You have to show that But this independent right that This component is not serrated component, right? It's not a point for example, but so in the open case. This is with fine So let me give you some hints of the proof So in some special case It's so let me give you some kids in some special case and I hope this clarifies Well, they can it was true. You will be considering like for example What would you be considering? Yeah, all complex co-compact actions of the group so Oh for for the surface. Oh, yeah. Yeah, so for a surface this definition is exactly a quasi-function space right quasi-function space is a quasi-symmetric embedding, right? Yeah, yeah, but I don't want to consider the bunch Yeah, yeah, yeah, yeah, the question function works fine and and the formula it comes from from Sullivan actually well actually from Bowen No, I can't oversee over city connected component because But or bad or good it depends on what you want So in some sense over see what happens is that there is this open state that corresponds to very nice geometric representations I'm the limit corresponds to other more interesting things, but Them itself. They are not geometric So let me give you some ideas big very big ideas of the proof Let me say that we're going to make it for K equal R and D equal 3 And gamma is going to be some free group. So let me let me let me kind of tell you what's going on here. I Think so so this kind of contains the whole This kind of in some sense contains the whole arguments at least for for 40 for number 4 K equal 1 here So understand this case you can understand everything so What is this? What is this? What is this guy? So here we are looking at this is a tool or Inside of SL3 are and So what is this guy in this case? So there's two ways of doing this so this you either preserve a plane and then a one that's not interesting So this is the reducible one. So what is this guy you can do it explicitly? So you can consider some poetic form in R3 Which is x square plus y squared minus x squared So this is from R3 R Right, and so you're gonna look at You're gonna look at the group of Linear transformation that preserve this in your form. So what's going on here? So This in your form where it has this has this light cone This is Q equals 0 which is a which is a cone because when you multiply by some number the number goes out And inside this cone you have this this convex set, which is what this is Q equal minus one I think So it turns out that when you restrict your form to the tension space of this subsurface This in your form is positive definite and so this gives you some remaining metric on this space on this surface So it turns out that this is actually constant curvature. So you kind of identify this That's another way and so very hard to identify with this robotic space Exactly like that. I mean you You restrict your form to the tension space to a point here and it's actually a positive definite Because okay because because you're best at them, right? and so This guy has to be preserved by this group So this guy is it's the somatic group of age two. So this I just said some inclusion But the other one is easy can be done. So this kind of gives It's a true R Say it's a morphic or say Up to finite index maybe This is kind of saying something like that and so When I projectivize this picture right when I projectivize this picture What do I get so this this cone It's gonna be some ellipse. It's gonna be a differentiable ellipse This is a projectivization of the cone right so this ellipse It's gonna be preserved by the whole group. Of course So this this is a regular ellipse. Let me put it this way because it's gonna be important So what I'm gonna do now, I'm gonna fix some free group. I'm gonna fix some convex co-compact action of some free group So this is so Q like this So what is this going to be? It's gonna be some short group set So this short group. So when you look at when you look at it in H2 Right the H2 picture of the group is that is this right? So I said it was going to be a free group. So this is an open surface no casps So the limit set is gonna be some counterset. So what I'm going to see on the other side is gonna be some So this image set is gonna be here It's gonna be some counter set inside this this ellipse this regular ellipse But I'm also gonna have just some counter set of lines of touch and lines Which are actually a teach a point in the counter set. I look at the line So I kind of have like two two maps So in R3 right in I have this this counter set right to the limit set So I still haven't before anything everything is containing this ellipse, right? This cantors this is going to be some counter set Inside Ellipse and when I look at the action in The two planes is going to be of this is going to be some also some counter set of the touch and lines, right? so When you move it a little bit When you move the representation now a little bit so you kind of move your gamma and you change this You move this and you change this So this is going to be kind of this is going to be preserved and this is going to be preserved Except the except this part because I don't have any tension anymore So let me just give a hint of let me just say something about that so After small the perturbation So there's gonna be some counter set that is preserved So this moves this kind of moves continuously. So this limit set move this limit sets And so So we're gonna have some counter set now around and Some kind of set of lines except that the tension doesn't touch it to anything because a Priority you don't know what's going on. It's just some lines So we just say a hint about this move why this moves continuously because it's related to this conference So the idea is that as Enrique explained Two days ago the dominated splitting is The bundles move continuously right So have a verisities again values move continuously and the main splitting is bundles move continuously So what what what's going to happen in this case? So this is some idea of La Burie actually He doesn't actually explain it in this in this form, but this is more or less what he's saying Just playing how this is continuously. So He's gonna come up with some linear co-cycle Right and this is your co-cycle is gonna have some dominated splitting And when you pick a point in this in this base space of this co-cycle and you look at the attracting Bundle of the domain splitting at this point. You look at this line It's gonna like here because it's gonna be one of these points So when you move your representation the co-cycle moves a little bit and so this domination is Preserved and so this this point is moving continuously. So this is very very very vaguely This idea of a Francois to actually explain Why is this kind of this kind of picture is preserved? So, but there is something more there is something more. So let me just put it like this This limit sets this limit sets Are the are the projective trace of the stable of the attracting? bundles of some co-cycle Having a dominated splitting So there are several ways to do this some more efficient than others depending on the purpose, so What we did with with with Haido and Raphael is also related to this to this this idea A few years ago, and so this is the reason the way they move continuously, right? So by now you have this Now you were starting with this ellipse, right? So this helps kind of give you some extra extractor So the ellipse gives you some extracts structure given by the by this ellipse So this is before the formation, right? So what is this extractor? So this is a compact set. So when you pick two points So let me Do it in green So when you pick two points any two points, what is this extractor? We say you pick sorry any three points So the fact that this is complex is saying that when I draw a line through two of two of them Doesn't contain the third guy so Given three points in this limit set This these things probably was distinct the line Between two of them. So these are two lines. I can take the the plane span by them So this is this line actually doesn't contain the third. So this is so This extra structure is saying one. So there's two things about this extra structure One that this is open. This is not too hard. So I was saying before that the limit is very continuously, right? So if x and y are far apart The land the line they span is is is continuous. The problem is that when they get close Right, you don't know what you have any continuity, but if they are far apart, then it's continuous And so and this is more continuously also. So if I move this a little bit This line is gonna move continuously and this point won't be in that line because it wasn't there before So this is this gets open when the points are far away But the action on triple so this is this is some hat trick in some sense so the action on triples is Properly discontinuous and co-compact. So this give you This allows you to always if you have two points that are close you can find some element that pulls pulls them away and Apply the argument so this this argument actually come from From Venezuela worry again this this kind of stuff this idea, baby So this is an open property And what is this saying? this extra Transversality saying that Okay Anytime I pick two points in my limit set. No now I have the form right and this is preserved So this is going to be true. So For the nearby so if I pick two points and then look the lander joins them It's gonna it's not much of a thing this third point. So these two points I can pick them very close And how to pick them and now that I know it's open the strong condition of them being being far It's just they have to be distinct So I'm gonna pick them close and this this line is far away from this line. So when I pick When I picked, I don't know some point here say and I look at this kind of Cocycle that I never actually explained Having this attracting point and this repelling point What I'm saying is that the span of two points nearby doesn't intersect the repeller the repelling bundle So when I I play my dynamics is going to convert to the attracting bundle So this is kind of saying that when I pick two points and I made an approach and make them both approach some third point This is going to convert to the red line to the tension So basically what you're saying is that What does this prove? I haven't say anything about how to the mission, but what does this prove? Is that okay at the beginning you started with some? Contour sets and limit set containing some ellipse so when you deform You don't you want to necessarily be inside that inside an ellipse anymore, but it's going to be some sort of C1 And this C1 control set is gonna you have to understand now how No, no, no you have a tension, right? So you have to understand the ellipses again the ellipses that I drew at the beginning And basically what you're saying is that when you you pick your element Your limit set is really is going to come inside the biggest So let me just say a comment that in this kind of situation for for Okay, well are and I don't know exactly more. I know I don't want to say more right now, but I know that funny castle has some sort of argument like this Also, but it's about convexity. It's not about this for any property that I didn't write this is what this is true funny castle and the chef dancing here and Also have some sort of convexity preserving bodies before the real case not in the complex case So I think I should stop now. Thank you And so it's like And so this gives you some if it isn't my conformality right there and you're gonna play it again, so leave on stuff Yeah, this is the idea Yes Can do is very quickly and this is something this is pretty much related to what we did with with shadow and So I said before right and we do have a convex co-compact Actual right a commercial co-compact action in h3 was in h3 Comments for compact action was that distance is here distance is here our course equivalent to trisic distances in the group, right? and This omitted splitting comes from the fact that when you look at this kind of representations that I have actually defined But the main property that's going to be around is that Instead of looking at distances in SLD. You're gonna look at the ratios of singular values This is this has to be comparable with the distance on the group on the on the group as an intrinsic This is so this is so this is accurate. This really comes from the work of what we did with This is kind of the main Metric property that gives you the domination of some oh because So so this is a closed surface, right? So the argument I gave so this is a closed surface So now the limit set is going to be the whole ellipse right and the argument I gave is stating that when you perturb a little bit It's still gonna be a c1 curve So what was actually very deep is this is a labor is theorem You can move far away and still gonna be a c1 curve and actually in this case. This is actually his arguments For small perturbations. Yeah. Yeah. Yeah. Yeah, you so the necessary. Yeah This is a three this goes far away. So, yeah Yeah, so this this proofs gives you that yes So it wasn't a saying like okay, so in the so in this specific case, which is d equals 3 and k equal r so This this c1 property holds You can deform as much as you want actually this this is a local argument And that's a that's a deep theorem. It's for this case the equal three is probably I don't know if it's joy goldman or maybe been one. Maybe maybe it's been all right joy goldman and then then for For k equal r and any d is la bogey