 Okay, I just switched on the microphone. So can you hear me fine from everywhere? Yes, okay And please tell me if I don't write big enough because I know it's a Okay, it's a big long board, but not very easy. I have to be careful about writing very wide Okay, so the title is it on lenses pseudo integrable billiards and generosity along curves So let me explain what I'm going to do. I'm going to present today two questions in mathematical physics one about the systems of eat on lenses and one about a special class of pseudo integrable billiards in ellipses that I will describe in a second and these are two mathematical physics questions Which appear quite different. They're kind of related to mathematical billiards and it turns out that the proofs of both The solution to these questions rely on the same result actually on two results and one Bossa results which concern Birkhoff's generosity and Oselle that's a generosity for certain curves in the space of a file Apices so it's an application of some homogeneous dynamics and in one case of some tech Mulder dynamics And probably I won't have time to explain at the end To some mathematical physics questions. So the outline I will start with these two applications I will explain you the problems and the result and then I will state the theorem in homogeneous dynamics Which is behind both of them and hopefully I will explain at least the first theorem how it follows from Homogeneous dynamics and I think if you've been to the end smart close course in the students were here There are some connections with what he might have taught last year. So I last week Okay, so let me start with the first application So I'm going to look at mathematical billiards. So let me first say what is a billiard in an ellipse, for example So I'm going to take an ellipse And then I'm going to look at billiard trajectories. So I'll change color. So I'm going to look at trajectories of Pointless ball which when hits the boundary of this ellipse is reflected So that the angle of incidence is equal to the angle of reflection So this is how a billiard trajectory bounces inside is elliptical billiard table. So when you hit You draw the tangent and then the angle here and here are the same And actually there's a nice way to draw this Trajectory is nicely. So if you find an elite co-focal ellipse with a larger table Which is my trajectory is not good. Okay, this is what a computer picture will be nicer So there is I'm drawing a red ellipse Which is supposed to be co-focal with the white table and it turns out that the trajectory which is tangent to this ellipse Billiard trajectory will stay tangent to this ellipse forever. So this is a good way to draw your billiard trajectory so Hope the picture is clear so a trajectory which stays tangent to this co-focal ellipse is actually a billiard trajectory So the angles with the tangents are always equal before and after So maybe I'm gonna write this so the billiard trajectories in an ellipse Stay Stays tangent to a caustic And I'm gonna write it in red since I've droid in red and The caustic is gonna be a curl I'm gonna call it see lambda and it's gonna become belong to the family of co-focal Conics, so I can write equation. So x square over and that's important a minus lambda plus y squared b minus lambda is equal to one Where lambda is in a zero a and here I'm assuming that the ellipse is the Conic corresponding to lambda is equal to zero. So x square over a plus a is a y square over b is One and a is greater than b. So this is your ellipse. This is the family of co-focal either ellipses of hyperbolas according to the value of lambda and Your caustic will belong to this family and this family is parametrized by the parameter lambda Okay, so we have a parameter lambda that parameterizes the family of caustics and In this example the billiard is integrable and I'm not going to define integrable systems. I guess here there are people who are experts Teach this so there are enough integral of motions and in particular what I want to say that There are some natural invariant sets so Let me call I lambda is gonna be the set of trajectories Which are the part of the face space whose trajectories are always tangent to a fixed Colic C lambda So this is invariant set It's an invariant set Which contains trajectories Tangent to C lambda so a fixed caustic and Clearly trajectories cannot be dense in the whole billiard But if you restrict to an invariant set in this integrable systems, it turns out that invariant sets are Isomorphic the flow on an invariant set are isomorphic to Floor as strictly to I lambda it behaves like the linear flow on a torus Flow on a genus one surface on a torus and hence in particular either For certain values of lambda you have periodic trajectories or if you don't have periodic trajectories Then your trajectory will be dense in the invariant set and uniformly distribute it in the invariant set But this is just classical case So what I want to look at today. It's not this well understood example But an example of what is called the pseudo integrable billiard obtained by adding a barrier to these ellipse so there is a recent paper which was published by Journal of modern dynamics actually one year ago, okay by Dragovic I'm not sure how to pronounce the names and Radnovic and Dragovic and Radovic in this paper describe a new class pseudo integrable billiards I will say in second what this means integrable billiards and This example I want to focus on is the following you take the same elliptical billiard But you add a barrier so maybe like this so for example ellipse with a barrier So now if I look at the invariant set I Still have invariant sets parametrized by the caustics But now the trajectory inside will when it hits the barrier the trajectory will Bounce at the barrier something like this again with the law of billiards Okay So this word pseudo integrable was actually could need it in the 80s by I think Michael Barry Who is in Bristol by the way? Sir Michael Barry and I should say and He means here the invariant sets I lambda are not genius one Surfaces anymore, but there are higher genus surfaces. So you get linear flown a higher genus Sorry the flown a higher genus surface, and then the natural question would be if our trajectories Billiard trajectories in this invariant set dance in in a lambda or Uniformly distributed for typical parameters for typical parameters lambda and the first remark of course there are some first remark maybe in the paper by Dragovich and Radnovich they show that there exists some exceptional lambdas. So I'm there exist Exceptional there exist lambdas for which clearly trajectories are Periodic, but there also exist the lambdas for which The flow the flow billiard floor restricted to this invariant set I lambda is Maybe let me say for billiard trajectories are Minimal so dense but not uniformly distributed are dense But not uniformly distributed so there are some areas where some regions where they spend more time than others Not uniformly distributed and it's natural to conjecture that those would be exceptional. So I measure zero set of lambdas and The first result I'm going to call here a one She's our first application of the year I might state later is indeed the following. So this is joint work Did I say that at the beginning what I'm talking about today? It's joint work which is tough front check who is in Turul, Poland and the wrong and she was currently the MSRI for last semester and So, let me say this is joint work with front check and She and myself So our first result is that for almost every Parameter lambda of writing the family of cal sticks the billiard flow flow on the invariant set is Minimal and uniquely ergodic And uniquely ergodic means in particular that all by infinite trajectories So all trajectories for example unless they hit for example the tip of the barrier for which we prefer not to define the trajectory So all by infinite trajectories are dense and equidistributed. So it's a positive answer to this dense and equidist Uniformly distributed With respect to the natural invariant measure for the billiard that I didn't mention, but sorry I lambda is this I caught it was supposed to be like a cursive. I Absolutely the same absolutely. Thank you It's supposed to be an eye in the script Fun, okay So it's the statement clear. So we have this family of billiards with barriers We fix a cow stick and we look at these trajectories with now bounce around the barrier and around this invariant Sorry The calc and the cut here. It's vertical and in the middle. Yes, and actually maybe I should say for every barrier So the barrier for every barrier the quantifiers are for every length of this barrier Sorry for every barrier in the middle vertical of every length and almost every caustic Okay, so that's the first theorem. So if you have any questions before I move further. Yes No, no, it's fine. Also. No, if you can one which is infinite in the future is gonna be yes but almost every trajectory would be by infinite and Yeah, yeah, I just want to avoid for example that they hit the steep and then yeah Okay So now I'm ready for the second application So this is an application to What we call systems of it and lenses. So let me start with what is an it on lens it so an it on lens So it's a circular lens. Hope you can think of it. You can realize it as an optical lens which acts as A perfect retro reflector It all ends it was discovered by it on in this century perfect retro reflector So you basically built a lens with a certain Fraction index which changes and it has the property that when I tried a light ray Optical of a ray enters your lens It comes out Parallel in in the opposite direction. So if the entering rate in direction theta it deflects inside the lenses So that when it comes out the exiting direction is minus theta Does it make the okay? So let me understand explain better what happens inside this lens here There is the center. So if you hit the lens with a light ray in a certain direction that you can draw the diameter perpendicular to tita and After being deflected by the lens here you get a rotation by 180 degree So this point is diametrically opposite to that and your Ray goes through this diametrically opposite point in parallel opposite direction, okay, and Again as I said these lenses can be realized and So what I'm going to look at I'm going to look at a periodic configuration of such lenses And you can see the picture already there. So I'm going to face a lattice lambda in R2 plane and radius R and I'm going to look at a lambda periodic array of Lenses of diam of radius radius R Okay, so this is a picture. So in this picture in This grid this yellow grid is my lattice lambda in the plane and then each Point in my lattice. I placed one of those round it on lenses and You should think of this. I don't know. It's okay Obstacles and then I'm looking and this is a periodic lambda periodic array of lenses of the same radius And then I'm going to look at the motion of a light ray inside this array So this is an example. Why this is a light ray in a certain direction By the way, this lens has this property whichever direction you shoot It will come out opposite and part of it, but here now if I fix a direction tita Then your trajectory will move alternatively in direction tita or minus tita. So And we will do something like this. It will snake around your system Okay, so remark trajectory each trajectory move each trajectory Move in something that I some Plus or minus theta direction Okay, so I can talk of the direction of a trajectory because up to plus minus sign. It's well-defined And now the question that you may ask so first of all, maybe I should say where does this system come from? So many people have heard maybe that mini course last week by Mark Lofen stromberson And you've heard maybe of the Lawrence gas. It's a billiard with hard balls So you can have I don't know if you answer mentioned that but you can have a variation of the Lawrence gas Which is like a soft billiard. So instead of having a hard ball You can put a cool on potential localize in the ball so that trajectories are deflected by the scuttlers And it turns out that for some special tuned value of this cologne potential you get exactly a needle lens so it's an exceptional case of a Special case of this cologne soft Lawrence gas And I think it was Jens Mark Lof with my colleague in Bristol who some day told me oh look This is the case that we cannot treat with stromberson because we don't have enough Hyperbolicity to understand the dynamics of this example and indeed it turns out that this is this system is Related to maybe say that again. It's related to a linear flow on an infinite periodic translation surface So it's actually a flat billiard and not a hyperbolic billiard. That's why it goes beyond the tools of stromberson Mark Lof Okay, the natural question in any case in this dynamics is our typical trajectories dance or not or is the system ergodic in the infinite measure sense or not and There was maybe like what where should I go? Maybe I Yeah, let's go from the beginning. I think it's okay. So there was a recent work by my co-author She's the front check together with Martin small Well, they actually discovered that if you do the medical stream simulations, you discover this quite quickly that the system is not at all Ergodic and that trajectories are not typically dense. So let me give a definition Usually what you see is the trapping phenomenon. So let me say a direction theta is trapped trapped if there exists Another angle phi which depends on theta such that every trajectory In direction t so we don't know interaction theta. So let me say parallel to theta. He is confined To a strip lie in a strip which has parallel to phi So I don't know if I can see it in the trajectory that I drew. I think it's too small But the picture is that okay You have this big array of lenses and you have in principle a lot of space in the plane to move on But what you can find is that there is some strip in the plane of finite width. So this is a strip Which has direction parallel to some direction phi and if you show that trajectory in direction theta this trajectory Okay, my picture it will bounce around but it will always belong to this strip Okay, so each trajectory stay in a finite strip and does not explore Ah Yes, sure. So for every trajectory there is a strip of body lines which contains the trajectory forever but all strips are parallel so that Direction does not depend that sure if I start if I start here, I will have another strip Yeah, but we saw so with and direction are independent The wisdom is related to the infinity norm of a transfer function for a co-boundary. So it's Actually, this is a good question. I don't I don't know the answer. I don't think so But no, I think it but that's actually that's a problem. We didn't analyze. So Yeah, but that I just concerned with the existence of these strips for now and let me say that what Franchek and small discovered is that this Trapping behavior. It's kind of typical for random lapses. So the result by Franchek Small is that Okay, you say that you fix a direction and we can say for simplicity that you look at vertical rays then so for every arm For almost every lattice Okay, well almost every here. There is a hard measure on the space of lattices for almost every lattice Or maybe I should have said that when you fix Lambda in an hour you always want to assume that there is no overlap So maybe I should actually say let me add this here Sorry, I should have said that the couple lambda are that you look at should be admissible in the sense That there are no overlaps. So you don't want to we want to be able to define the systems So you want to choose an arse so that the balls are this joint given your love But if you fix are there is an open set of lattices which are admissible And there is the hard measure on this set and almost every random lattice in that set We'll have the property that the vertical trajectory is bound is trapped Vertical direction Is trapped So this is a nice result But as you know in physics you don't like maybe a random things because then you ask yourself Well, can I produce an example and actually in the paper they also show a specific example So they give you some lattice and some Direction which is trapped but nevertheless their natural quantifier from my point of view that that was bothering Front-check asking him about is what do can we do something for all lambda and this is indeed the Sorry vertical nothing is special so vertical is just you rotate your picture so that you put it vertical It's just to say the idea that this quantifiers work fixing a direction and throw your lattice at random It's just a rotation Nothing is question No, it's actually Actually, I think she just visited Bristol at some point and he had a discussion with Jens Markloff and Jens was convincing me that It's nice to do this, but I still don't like the quantifiers So I wasn't happy and apparently the rest of the paper by Shistoff also asked him well, what about the lattice so we saw started thinking of this question and Okay, so the theorem the second application I'm going to call it your ma2 This is again, I'm not gonna rewrite Shistoff front-check she and myself Is that now you look at every for every R and for every lambda such that the pair Lambda R is admissible so whenever you can draw this picture of a ray and For almost every direction so again, it's almost every here Theta is trapped so again, this is the result for random lattices and here instead is a result for a fixed Latis and the result is still true for almost every direction. Okay So this is the second result Again, now, I think I'll move to the homogeneous dynamics result behind both these applications So if you have any question on it on right now, is it clear the picture? so now I want to Set up some notation and I think some student told me that they saw last week the space of a fine lattices ASL to R over ASL to Z. So that's the setup where I want to work now So I will recall the notation for people weren't here or who doesn't know already. So let me Now I'm moving to the I'm gonna state now a homogeneous dynamics result and maybe at the end I state also Tristation surface result, but okay, so set up and I'll explain the connection at least with application one and So the setup is the following. So you look at ASL to R. So this is SL to R This is the space of two by two matrices with the terminate one real entries Semi direct product with R2 and don't get scared because I'm gonna react in a second explicitly what this is in terms of matrices. So this space lives in SL3R. So it lives in the space of three by three matrices and An element here You can think of it as a pair A Xe where A is a matrix in SL2R and Xe is a vector in R2 and You can embed this in R3 by putting your two by two matrix here So here you have the matrix A and here you put Xe1 and Xe2 and Here 0 0 and 1 So you can look essentially at the three by three matrices which have a matrix here and vector here And as you the product multiplication on matrices will give you this semi direct product structure between pairs okay and And then you can consider a lattice inside ASL to R and the basic example Is take the same matrices where the entries now would be integers for example ASL to Z and then I want to consider the quotient so the quotient of ASL to R Modular ASL to Z and This quotient is the space of affine lattices. Let me say that and explain that in a second This is you might have seen the space of lattices on the module lattices You certainly see it so if you were here last week This will be the quotient of SL to R module or SL to Z when I put this ASL this become affine lattices So let me draw an example so So affine lattices means a lattice but which is shifted so that the origin is not at 0 0, but it's at other point So you can single-facing lattice lambda and lambda will be if you have an a pair A XE you will take as lattice a Applied to Z2 to the standard that is this is a lattice and then you look at lambda plus XE work see See is a vector in R2, and you are gonna consider XE as an element in R2 mod lambda So if you shift by a there's a coset of XE then you get a new lattice Which is not centered at the origin, but it's centered now. I think my axis is here. So it's It's a shifted lattice, okay, and There is a dynamical system on this space which is given by the diagonal flow of The following form so you can take a matrix e to the t e to the minus t 1 the rest is 0 and This acts Quotient e to the right so this acts by left multiplication on your space And this is the diagonal flow, which in this case is also the geodesic flow okay, geodesic or Geodesic flow on the space of lattice. Okay. Let me say just a diagonal flow So this is the dynamical system and Maybe here is fine Maybe it was nice to keep the picture. Let me keep the picture and move to the other side Okay, so we have the space of a philatesis. We have a diagonal flow and This flow is ergodic. If you know what ergodic means. This is a very standard Resulting homogeneous dynamics, but let me say if you don't know what is a godly Let me give you more precise definition. Let me say that a point in my space acts is Birkhoff generic So Birkhoff here is the Birkhoff of Birkhoff ergodic theorem. It's Birkhoff generic if Essentially the conclusion of the Birkhoff ergodic theorem holds so if take any function which is continuous maybe even come back supported on x and then look at the ergodic so take your function and Compute it along the orbit of x a t of x and Integrate from 0 to capital T in time So this is the ergodic average of f along the diagram flow and the point the partner starting point is Birkhoff generic if this stands as T grows to the integral of F on the space With respect to the natural hard measure on the space of lattices so 80 ergodic If you want you can take this as a definition implies that almost every hard almost every mark I knew almost every Point is Birkhoff generic. Okay, so this is just ergodic theory For the space of lattices So now I'm not gonna I'm going to look at a special curve a class of curves in this space of lattices a family of one Parameter family of lattices. So if I look at the curve Even if I know that almost every lattice is Birkhoff generic the curve is that Small set it's a measure zero set So I don't know if every point in the almost every point on my curve will be Birkhoff generic so the first result that we have is indeed a result about Birkhoff Genericity along curves from here the title Genericity along curves and So So this is theorem. I'm gonna call it b1, but I'm never sure if I'm gonna state b2 so this is front check she and would she cry and The result is that why look at this some special curve which have the following Form so I'm going to call them gamma of lambda and I'm gonna write them in this three by three Matrix, so I'm going to look at class Of the following form One s Zero one, so this would one and then here I put a function field as Okay, this is especially this is these are called sometimes Horo cycle lifts because they occur which lifts horo cycles and where From zero one to R say it's a C2 function And it's not degenerate Non-degenerate means that for every rational line a Curve is not degenerate if it's not it doesn't come any positive measure subset is not contained in a rational line so I'm looking at Let me write Fee here. It's a curve which is not degenerate So if I fix a rational line the Lebesgue measure of lambda such that lambda phi of lambda Is contained in this line is zero So it's a curve which is not flat and it's not that tangent is not contained on in any Rational line. Okay, so this is a family of curves. You should think as lambda changes. Oh, sorry. Sorry I'm confused everybody because I change parameter. This is meant to be lambda Okay, sorry. So this is a family of Points in my space of a phyla species parameterized by lambda and as lambda change I get a family of a phyla species and I'd like to know that the result is about Birkhoff generosity along families of this form Actually, there is a much more general family of curves that can be treated But I want to state only the simpler result because it's I don't want to state more Sophisticated on the generosity condition. So you take take any curve any Gamma Such that it has this form where phi is not degenerate Then the result is that for almost every lambda The point to gamma lambda on this curve is Birkhoff generic So the picture is that you have this curve gamma in the space of lattices Parameterized by lambda and if you pick at random one point on this curve and flow it under your Diagonal flow it will be Distributed and Birkhoff generic, okay And now maybe a remark some people in the audience I don't know if anybody some for sure have I have heard of a paper by Elkis and McMullen It's a paper on gap distributions of fractional parts of square root of n So at this and McMullen look at the problem, which is kind of similar but in reality not connected So so they have a following result. They look actually at the same type of Horticycle diffs So they have a curved gamma in the space of a phyla species with the same exact assumptions But they look at the different problem So they take this curve and they flow it as a curve. They push it by 80 and they prove that 80 gamma equidistributes so So if you put some uniform measure on On this curve and push it by the diagonal flow it will spread uniformly So and what I want to remark is this is these two problems are not connected. So neither implies the other so Okay, that was just by the way We also have our so this theorem B1 as I will try to explain is behind application one and also application two But we also have a third application Which is about the gap distribution So we have a different result about gap distribution which can be gathered by these and if you are curious You can ask me at the end because I don't want to go into that now Is this does it make are you still? With me, okay, so I would like to sketch now why this application This result gives the application billion seen ellipses with barriers And for the second result of it. Yes Yes, yes. Yes, absolutely. Yeah, this is unstable. You're right This is like a one curve in a two-dimensional unstable Yes, absolutely, but okay Okay, so I don't want to sketch how Theorem B1 implies theorem a1 a1 and We'll be brief, but I just want to give you some idea So, okay, the main point that we start with an ellipse with the barrier and we look at this orbit Billiard trajectories in this ellipse tangent to a given house So the first step was done in the paper by Dragovich and Rod Novich and it's essentially the main observation to me that comes from our paper The key point in their paper is that if you know integrable systems, you know that you can change coordinates using the right elliptic integrals and You can change coordinates change of coordinates by elliptic integrals and You can map this maybe here. I'm fixing an invariant set lambda. You can map this invariant set into Polygonal billiard, so you can map your maybe I should go on to the kufocal you can make this picture into a rectangle and the barrier becomes again a barrier and what used to be a Billiard trajectory there becomes also a billiard trajectory actually in my picture is not exactly a rectangle, but it's a Strip so these two sites are glued while these other sites are Reflect so here you have a trajectory which Does the following? Here is reflected and so on and the key point that actually this trajectory is in direction No matter what caustic you start from this angle. It's always pi over 4 So this is always 45 degrees slope Okay, so the first reduction which is by elliptic change of coordinates is the invariant the flow in the variant set maps to a flow in a Polygonal billiard in this case a cylinder with a barrier, but the key point is that Okay, maybe that would be remark that for every lambda Here the billiard Let me call this system be lambda. This is the billiard which depends on lambda for every lambda the billiard trajectory in the lambda is Has Theta which is in the set of pi minus It has no plus minus one So it's either plus minus pi over four or plus minus three pi over four But the billiard table actually change with lambda So you have as you move caustic here you have a family of billiards for which The direction is always fixed, but the table change So some people may have heard that there's I mean my my background is from tech Muller dynamics and the study of rational Billiards in polygons and there is a celebrated theorem by Kerk of Maison and Smiley from the 80s Where they show that if you fix a table for almost every angle the billiard flow if you fix a table with Rational angles in particular for example this table and you change the angle for almost every Direction the billiard would be uniquely ergodic and trajectories would be dense a uniform and distributed so they keep challenge here is that I'm not fixing the table and I'm not Changing the direction as the natural parameter in the elliptic system as the caustic change The table is changing. So I have a one parameter family of billiard tables with the fixed direction And this is like a curve in the space of billiards. Okay, and now I want to know that the typical billiard in this family in that direction has Uniformly distributed trajectories and this is exactly maybe there is one more step that you can do You can and again if you were students who were here last week When you have a billiard, you can unfold it which means you can take in this case four copies of your original billiard table one per direction I don't want to explain this now, but in this case you can take four copies of this thing and what you get is that your billiard trajectory in these four copies is straightened to a Linear trajectory. So here the direction is fixed to be pi over four And now you reduce yourself to what is called This is gonna be when you glue these two copies according to natural identification in this case You glue this with this this with that and in this lit and this barrier you want to glow Glue this iron This side of the barrier with that side of the other barrier and the other side of this barrier You want to glue it with that barrier I want to claim that you get a surface and a surface of genus 2 and you can see it because each of these two Rectangles glue into a torus and this Barriers you should think of them as a slit in the torus. So you cut open a slit in the torus And then you glue these slits and this will form a genus 2 surface So, okay, moral of the story change coordinates you get the billiard you unfold your billiard and You get a surface of genus 2 with the linear flow on it So this basically each invariant set is a linear flow on this genus 2 surface That's how you see that also invariant sets are higher dimensional surfaces and now this Surface is actually of a special form this surface is actually this is a flat surface Which is actually a double cover of a torus, which is a double cover of a torus with a slit It's a double cover if there are two tori each with a slit and each of these two tori Is in a space of a five lattices so you can think of each torus as a lattice and each slit as a Vector So moral of the story and I think I'm running out of time, but I would like to still state one Theorem be to without explanation. So let me be quick now so So I have as I change the caustic I have a family of billiards and as I change it as I unfold this billiard. I have a family of surfaces so theorem theorem theorem theorem be to Be one sorry implies that for almost every lambda for almost every lambda s lambda is Birkhoff generic and Actually here I'm with it which is where I'm going to lie for the teichmuller flow So this diagonal flow on the space of a five lattices Lift to the teichmuller flow on the space of transportation surfaces that probably you don't know what it is But in which case I apologize But this result I claim it's enough by some well-known result in teichmuller dynamics by measures criteria for a good city This implies what we want that for almost every for almost every lambda I lambda the invariant set has dense Okay, that's dense trajectories dense uniformly distributed trajectories it has uniquely ergodic flow Maybe let me write it as a uniquely ergodic flow On it Okay, so it was brief, but hopefully you get the sense of why this Birkhoff genericity Comes into play and maybe I just want to state Another genericity result which together with this one enters in the So I conclude with the statement of theorem 2 and it's also gonna be more for experts So the second genericity result, which is used only in application to maybe no from the authors now So it's the following so you look at this time you look at the family of curves Let me just write it like this. It's a family. It's a curve in this space of surfaces of genus 2 Where? As lambda here are actually Double covers it's a little bit more general, but in this special case. I want to just say it's double covers of a flat tour of torus with a slit So I'm looking at Special family of flat surfaces each of which is composed by two copies of a torus with a slit look together with the state and and I'm a little bit big here and look at the curve of this genus 2 surfaces which project to a Curve as in theorem 1 So I have a curve of surfaces of genus 2, but I want that as I move my curve the torus with the marked point with a slit moves in the space of a fine lattices as one of these Horocycle lifts that I described in theorem b1 and then This is again where I'm gonna lie because jolly experts will In this space of flat surfaces, there is what is called the concierge-zorich Co-cycle there is a linear co-cycle which consists of taking product of matrices along the Tecumuller flow and I just want to state it because I cannot define this So what we are proving. Oh, sorry, and there is lambda for almost every lambda this surface as lambda I want to say it's Ozele that's generic So Ozele that's Multiplicative ergodic theorem guarantees the existence of Lyapunov exponents for this linear co-cycle So for almost every lambda this in this family, we have Ozele that's a genericity for the concierge-zorich co-cycle. I Don't know if probably maybe Anton Zorich if you were here gave it up Did Duncan Zorich mention the concierge-zorich co-cycle any student who was here? Yes, okay So again, this is another result about genericity not Birkhoff genericity, but Ozele that's a genericity for The product matrices. Yes, and this is what we need to show the existence of these bands because these bands are related to Eigen direction of certain Lyapunov exponents and they exist exactly when your surface is Birkhoff Ozele that's a generic. Thanks a lot