 OK, zelo povedaj. To je pravda, da imamo Korina Učigaraj, kaj imamo povedaj, kaj je zelo povedaj, in Lagrangi Spektra. Prečo, prečo, povedaj, da se povedaj, da je na webače google doc, document, kaj vsi vse povedaj, da se povedaj, nekaj zelo, nekaj se povedaj, je to, da se da vsegače. Na sada je, da je bilo v tomom, da se prišli početno, in bomo razgledati, da se pače. Če vidim, da se tače, da se prišli početno, da se početno, da se prišli početno, da se prišli početno, da se prišli početno, So, as some of you know, I'm actually originally from Trieste, so it's a great, even though I left when I was 18, so before I had scientific contacts with the city and the institutions of the city. So it's a great pleasure for me to be back in Trieste and give a talk in my home city. So I've been to some of the dynamical system conferences that have been given here, so what I do is more dynamics than geometric theory, but I try to find some topic which has more geometry and some groups in the talk. Ok, so first I want to remark that the work that I'm going to talk today about two works, which are joint works. The first is a joint work with Diana Davies and Irena Pasquinedi, and this is a work on cutting sequences on which surfaces. I'll tell you what this means in a second. And the second is a joint work with Mauro Artigiani and Luca Marchese. So I'm going to put the names here, I should have done it earlier. So these two works, both are works on which surfaces, on dynamical properties of which surfaces, both of which involve the study of the which group action on the space of affine deformations of a which surface. So that's why I'm going to talk about them together. And the outline, I will spend some time at the beginning because this is not a common background for this type of audience. I'll spend some time about basics on what is a which surface and which group. Then I'll introduce the first result on cutting sequences. I'll introduce and motivate the second result on Lagrange spectra, which are also in the title I had penetration spectra. It's actually the same thing, Lagrange penetration spectra. And then I'll try to give you some ideas of the proofs of the two results. So you will be back and forth between the two topics and you have several locations to wake up and catch up if you got lost in some part. OK, so time to do some background. So I know that the students who were here last week had a course with Pascal Ubert. So you're probably all well-prepared on what our translation surface is. So actually the best description for me is the easiest. So you take a polygon which has pairs of parallel sides, isometric parallel sides, glued together like A with A, B with B. So in this case we have a surface of genus 2 glued out to a regular octagon. So this is the regular octagon surface. And whenever you get a surface out of gluing polygons, your surface naturally carries a flat metric. So apart from possibly finitely many points which come from the corners of your polygons. And this is the set of singularities, sigma. So outside is finitely many points. In my case all the vertices of the octagon are a unique point on the surface. Outside this point you have, other point has a locally Euclidean neighborhood and changes so coordinates in your manifold definition are given by translations. In complex coordinates z goes to z plus c. So this is why we call those translation surfaces. And the singularities, the special points, we have an excess of curvature because we are on the surface of higher genus so there should be some negative curvature. And these points are conical singularities called, what the model if you want is z goes to z to the k. So it's a k cover of the plane. So on translation surfaces there is a well-defined notion of lines. So the geodesics are flat lines and you can talk about directions. So I can say there is a line in direction theta. Direction is well-defined. Apart from the singular point. At the singularity, so what's happening? I'm right too far, yes. At the singularity there could be many lines in the same direction which hit this point. So this point you can think of it as a saddle on your surface. So these are how linear trajectory looks on the surface. And again we are thinking of surfaces not polygons. So two polygonal presentations of the same surface are two polygonal presentations are equivalent in the same surface if you can cut and paste your polygons by translations. For example, in my picture you see you can take this surface and cut and paste the polygon and you will get again the same flat metric on the same surface of genomes too. OK? OK. So if we have two translation surfaces we can look at affine difiumorphisms among them. OK. Maybe I should have said that affine... I missed something. It's a continuous map. It's a homeomorphism which actually maps singularities to singularities. And it's affine in each chart. So it's a linear. It has a linear part plus constants. And the linear part is independent on the point. And I can look at self-affined maps from a surface to itself. Affine automorphisms of S. And the group... This is a group of affine automorphisms. And the which group... I'm too far. The which group is the group of linear parts. So I look at these affine maps and they look just at the linear parts. So a linear part will be given by a matrix. And this matrix is... Affine difium... automorphisms have to be area preserving. I'm not assuming necessarily they are orientation preserving. It's not standard. So there are actually matrices in SL2R plus or minus. So 2x2 matrices will determine plus or minus 1. Possibly I can also use reflections. OK? So let me give you an example. So if you take the square torus like here I claim that the matrix 1101 is in the which group. So if I apply 1101 that's actually a shear. Or if you want it's a full then twist in the square. But after I shear my square I can actually cut and paste it back to square square. Parallelogram can be cut and pasted back to a square. So this shows that this gives me a self-automorphism of the torus into the torus with derivative 1101. Here's a more sophisticated example. So if you take the regular octagon I claim that this matrix is in the which group. So this is also a shear where here the upper entry is here. So if I apply to my octagon this matrix what I get is a shear octagon. It's this black figure here. And this is a shear octagon and I drew it inside a grid made by pieces of the original octagon. Because I want to show you that like in a jigsaw puzzle you can rearrange these pieces of the shear octagon back into the standard octagon. I'm wondering whether Pascal Ubers show this example in the course. So this matrix gives you a fine automorphism. Are you shear and then cut and paste back? So we want to focus today on gravitational surfaces which have plenty of self-affined automorphisms. Those are vich surfaces. So a surface we say that it's a vich surface or also lattice surface if the vich group is a lattice in SL2R. So again same two examples. If you take the torus you can add to this you can either add the other then twist, vertical then twist so this is actually a rotation and you get the full SL2Z which is a lattice in SL2R. And for the octagon you can take for example, you can take here I took two, I like these reflections which are determinant one. This is a horizontal reflection this is a slanted reflection so all isometries of your polygon will in particular give you a fine self-automorphism. Izometries and then I can put this non-trivial element that we saw together and this will generate a lattice in SL2Z. So the regular octagon surface is a vich surface and this one. So this is a little bit of introduction because it motivates people. People have, like molar dynamics are very interested in vich surfaces because they are interesting from the dynamical point of view that it is via dichotomy that all trajectories are either by infinite trajectories are either periodic or dense and uniform and distributed. I don't want to spend time this is just their motivation. And also if you look at the SL2R action they generate closed orbits known as tec molar curves. I'll tell you more later on in the proof part. And people have tried to list, classify vich surfaces so we saw that the square and the octagon and more in general any regular polygon will give you vich surface and this is a seminar paper by vich where he defined vich groups and discovered vich surfaces. There are some examples which were found by Ward and then a little bit more than 10 years ago there were two infinite families of vich surfaces which were discovered one by Boh and Moller by Boh and Moller this is the Boh-Moller family and another family was discovered independently by Calta and McMullen it's a family of L-shaped vich surfaces in genus 2 and the Boh-Moller surfaces I will focus on them today so I will tell you more about them and then Mokamel gave a talk on Tuesday here I think he didn't say he talked about the gothic locus and in the gothic locus there are infinitely many new vich surfaces which have a cathedral shape and there is lots of research and people working on proving that there are infinitely many with fixed complexity genus and singularities as I said apart from these classical examples the Boh-Moller surfaces have vich groups which are triangle groups while these L-shaped family surfaces have more complicated vich groups as I guess Ronin has studied so I want to focus on the Boh-Moller after regular polygons they are the nicest family and they are obtained by gluing a collection of polygons which are semi-regular so this is all these polygons together glued according to the numbering that you see one with one six with six this is one surface and this is another surface and let me first say pat Hooper polygonal presentation so Hooper discover this simple polygonal presentation of these vich surfaces so first of all vich surfaces Boh-Moller surfaces are parameterized by two indices M and N so the SMN surface the MN Boh-Moller surface is made by M semi-regular polygons each of which has a symmetry of order 2N so this is S4-3 so there are four polygons which have order 6 symmetry so these are semi-regular hexagons so the angles are equal but the sides come in two lengths and these are triangles but they have still order 6 symmetry and I order 3 I think I said it wrong and this is S3-4 where there are three polygons with order 4 symmetry and you have to pick the lengths specifically so they are not any two semi-regular you have to pick them carefully because you want the surfaces to have a global shear like the one in the octagon that I showed you before and maybe I could have said about the octagon that this shear you can think of it as a simultaneous then twist so the octagon has two cylinders and you are doing a then twist in each but the cylinder modular are commensurate so you can do this to simultaneously and here you have cylinders and you want to be able to do a simultaneous then twist in each cylinder to get a globally defined shear on the surface ok so this is as much as I wanted as an introduction I got too far and then I cannot click ok and I want to close maybe ok so now it's the first result I want to talk about it's about cutting sequences so we have a Vich surface and you want to first of all give names to the pairs of sides which are glued for example in the octagon I am using the alphabet A, B, C, D to label the sides and if I have a bi-infinite linear trajectory what am I doing wrong with this clicker I think ok this is an example of a bi-infinite linear trajectory you go out and come back and I want to associate to a linear trajectory a sequence of symbols to kind of give a symbolic coding of the trajectory and this is a sequence in the alphabet A, B, C, D called cutting sequence and the idea is just that you look at the sequence of sides that you hit as you travel along your line so in my example I hit A so I record an A then I hit a B then I hit another B then I hit a C so that's the cutting sequence of a trajectory and I only want to look at bi-infinite trajectories that give me bi-infinite cutting sequences so the problem I want to talk about is how do you characterize these cutting sequences among all sequences in the alphabet and what do I mean maybe I should say that cutting sequences are rare I mean if you flip a coin actually you cannot flip a coin with four letters so if you draw a tetrahedron dies and at random you will get random A, B, C, D but here the sequence that you get is very far from a random sequence because it has a lot of structure and it has actually very low complexity compared to a random sequence so if I give you a sequence how can I tell if it is a cutting sequence and how can I build cutting sequences by drawing the lines by hand and really have issues with this ok, so just to say that this is a problem which in the case of the torus is very classical because in the case of cutting sequences for the torus are actually well first of all you can think of them as just how a line hits sides of a square grid where you record by A and B horizontal and vertical sides and these cutting sequences are called Sturmian sequences they have been studied by headlong and Morse more than a century ago and they have the lowest possible complexity among non-periodic sequences I don't want to tell you what that means so apart from Sturmian sequences there are nice references for their characterization there is one way to see geometrically by Caroline Siris I strongly recommend this paper on mass intelligence and I give it to many of my undergraduate students because it is beautifully written or there is another characterization more combinatorial by Arnou that I will tell you about today and then a few years ago in joint work with John Smiley we managed to give a similar characterization for octagon cutting sequences and two endgones, polygones with even number of sides inspired by this series work and then also Ferenci characterized the language of this cutting sequences for regular polygones and after our work there was kind of a surge in interest so Diana Davis and Diana Davis with daughters they looked first at the pentagon and then at octgones so you have to take two polygones with odd number of sides to glue them together so double octgones and found that our kind of characterization goes through and there were some other works Diana Davis had some partial results on this bone-molder family so what I'm going to talk about the result that I'm going to talk about is that it's about the bone-molder family so which is in some sense the next step up from regular polygones because for regular polygones which group is a triangle group and for bone-molder you have a triangle group with two elliptic points so I will see that later and I think it gives a fairly good intuition of what would be the general picture so the substitutions these characterizations will be in terms of substitutions and what I call esadic systems that's the language that people in work combinatorics like and especially people in Marseille like Karnu also Valerij Bertha ok so let me tell you what the substitution maybe you've seen in your life I don't know so it's a substitution on an alphabet A it's a map which to each letter of the alphabet associates a word so for example these are the two Sturbian substitutions sigma 0 and sigma 1 so sigma 0 maps A to A and B to AB so to each letter you associate a word and this is sigma 1 and substitution then the action extends in words or sequences by just a position so if I want to find sigma 0 of sorry this should be AB I change 0 1 to AB so sigma 0 of AB is sigma 0 of A and sigma 0 of B so it's A AB and so on so you just apply to each letter to each letter you substitute a word that's another example applying sigma 0 squared and then applying sigma 0 squared to sigma 1 cube and by applying sigma 0 actually this is a piece of a Sturbian sequence ok so people often look at sequences which are fixed by a substitution so sigma of omega is equal to omega but if you have a word which is a fixed point of a substitution you can actually produce it as a limit by taking some letters you have to choose the initial letters suitably and applying the substitution many, many times so as you apply the substitution more and more times you get longer and longer blocks sigma to the N of L and you can imagine if this longer and longer blocks stabilize and converge the limit sequence you can converge towards the limit and you call this kind of the what do you mean here I am just talking about substitutions so there is a linear flow I am going to study maybe I am going to far ahead let's talk about it ok but I don't want to talk about fixed points of substitution but I want to talk about sequences generated by finitely many substitutions so these are acidic so as I take s to be a finite set of substitutions and then look at products of this finitely many substitutions so you can look at limits where I take some pick from this set substitutions and apply them and then I get longer and longer blocks I can apply however I want my substitutions and if I get something which stabilizes and actually I do if you grow your sequences towards the middle so you you will get sequences which actually stabilize and converge towards the limit so if a word has a limit can be obtained by longer and longer products of these set s substitutions I say that the sequence has a acidic expansion so acidic expansion so sturmian sequences can be characterized by acidic expansions so a word is a sturmian word or it's a cutting sequence of the square actually I should say it's in the closure so we always characterize the closure of the set of cutting sequences if there exists a sequence of integers such that the word infinite sequence has an acidic expansion and here I'm alternating sigma 0 and sigma 1 sigma 0 and sigma 1 are the sturmian substitutions that I showed you before and I alternate first I apply sigma 0 a 0 times then again sigma 0 a 2 times and so on and this is if I can write every sequence as a with an acidic expansion I say that this is an acidic characterization and I should say this word acidic was due to ferensi and it reminds of Varshik and this ak actually have a meaning dynamically they turn out to be the continuous fraction entries of the slope of your line in the plane and they can recognize them combinatorially ok so this is the type of characterizations I'm gonna look for and as I said the first main result is about bone moller cutting sequences so if you take a bone moller surface and label the sides in this joint work with Diana and Irene we show that you can actually write n-1 times n-1 substitutions and we have a recipe to actually produce them so that a word infinite sequences in the closure of the cutting sequences on the bone moller surface if and only if there are in the actually this I parameterize these substitutions on two indices for reasons that maybe we'll be here later and if I have an acidic expansion with the substitutions so this is an acidic characterization and you can use it to actually have a recognition algorithm to know if a finite block is actually a part of a cutting sequence or not and you can recover this indices can be recovered combinatorially and you can interpret them in a continued fraction generalized continued fraction ok so this was the first result I hope you are still with me so you can wake up if you are lost because there is a new topic so which is the second result on which surfaces and then we'll go through the tools in the proofs so this is the Lagrange spectrum part so are a generalization of a classical object in number theory but I'll first give you the general definition that I want to use and then tell you what it is in the classical case so it's again a geometric object for me so I take a translation surface and saddle connections and the translation surfaces are lines which go from a saddle to a saddle from a singularity to singularity without hitting singularities in the middle and if I have a saddle connection I can represent it in the plane like a segment and the displacement vector or holonomy vector is just the difference between the end point so it's really a this vector in the plane so this is the holonomy of a saddle connection or displacement so if I fix a direction on my translation surface and I have a saddle connection I want to define one more thing the area of saddle connection in that direction so I'm gonna take this area to be an area of a rectangle which has the saddle connection as a diagonal and the sides are parallel to theta and theta orthogonal so this area is the area of the saddle connection in the direction theta the light blue area and if you want you can rotate the direction to make it vertical and then it's real time imaginary part ok so what is the Lagrange spectrum so if I fix a direction theta I want to look at saddle connections which are more and more in the direction theta and look at the areas of the saddle connections and take the lean soup of one over the area so maybe I have one better picture so as I take saddle connections which are better and better approximations and look at the rectangles which I can inscribe and take these areas and I want to take the smallest areas as I grow so to make the lean soup big I need the area of one over I need the area to be small so I'm searching for saddle connections in that direction with small area and this is the Lagrange value in the direction theta the Lagrange spectrum is the collection of Lagrange values as theta changes so it's going to be a set of numbers yes of course I should have said that so these rectangles when I draw a rectangle I mean so there are no actual flat rectangles so there are no other singularities but actually actually maybe you don't need to require it here but in reality the saddle connections which minimize the area are the ones which come with an embedded rectangle ok ok, so there is a more general definition for a saddle to are invariant log chi of translation surfaces that we gave in joint work with Kuber and MacCazer but no this is a special case and this is a generalization of the classical Lagrange spectrum so if you take your surface to be the square the unit square surface then it turns out that this is a well studied object which has also the definition in terms of dieofantine approximation and that's maybe the you can look at denominations of continuous fractions there is a dieofantine approximation and there is also a geometric meaning in terms of penetration spectrum so if you look at the modular surface with hyperbolic geodesic flow actually you can take a ray geodesic ray in a certain direction theta and you can look at excartions into the casp so you want to know how far you are going in the casp and you can measure excartions through a proper function which goes to infinity in the casp and I want to use the saddle connection flat system so this is the shortest length of a saddle connection on the torus and these values can also be computed as the lim sup of 2 over sisto squared along my tec molar ray so I look at 2 over sisto squared so remark actually now that the typical geodesic will go arbitrarily far in the casp so the system will go to zero and this number will go to infinity so this value is actually infinity for a typical geodesic the only geodesics for which it's finite are bounded geodesics or closed geodesics so closed geodesics stay in the casp and then what you are measuring is the maximal asymptotic excartion so you are kind of measuring the diameter of the limit set of your geodesic so this is why this spectrum is called the penetration spectrum ok this is, if you didn't get it, it's ok it's just a motivation about the classical case so what is known about the classical spectrum this classical Lagrange spectrum can be studied for more than a century so we said it's a subset of real line union infinity and for most values it's actually infinity but people, I mean it's closed and it's the closure of quadratic irrationals which correspond to closed geodesics on the modular surface and the minimum it's square root of 5 it's called Hurwitz constant so there is a minimum it's my blue dot to the left in my line real line minimum is there and the beginning up to 3 there are discrete values so those are called Markov numbers then that's what I want to care about today from some point on this Lagrange spectrum contains a full interval so from a value actually the best value is the Friedmann constant from here on the full semi line is contained in the Lagrange spectrum and what about in between the discrete part and the interval interval is called Hall Ray and in between the structure is quite complicated it's like a counter like structure and there are recent results by Moreira on the household dimension so this is the picture of the classical Markov spectrum discrete beginning, Hall Ray and complicated middle so there were many generalizations of Lagrange spectrum other fuchsian groups higher dimensional hyperbolic many folds, bianchi groups by many people and for translation surfaces some properties we proved in this first paper with Kuber and Marquesa and for example the Hurwitz constant was recently announced by Bošanica and Delacroix so what I want to talk about today is the Hall Ray so what we proved with Mauro, Artigiani and Luca Marquesa is that this Hall Ray phenomenon persists for any Vich surface so take any Vich surface look at the Lagrange spectrum then the Lagrange spectrum of any Vich surface will contain a half ray so there will be an R from which all values are achieved actually the proof uses Vich groups and arguments by Hall and symbolic coding but I hope to get there a little bit later ok, so now I finish the statements part so we have a result about cutting sequences on Vich surfaces especially Bo-Moller surfaces and result about Lagrange spectrum ok so now I want to give you some tools from the proofs any questions now? yes? so the so actually I am kind of lying because there is something called Markov spectrum which is similar to the Lagrange spectrum but instead of soup you take a soup and people have studied the Markov spectrum for other Fuchsian groups and there is actually maybe I should have said this, I should really say so there is a result by Schergon and Schwitt where they prove the Hall Ray for Markov spectra but there is quite a big difference ok, there are different definitions so say that you want something which has casps so if you want to talk about penetration spectra but ok, if you have a Vich surface you always have casps but yes so for this Markov spectra there are results about the Vich but in some sense if you want to achieve a soup it's easier than if you want to achieve something which gets to that value infinitely often so this actually creates some confusion so they say they tried to understand Hall Ray for Lagrange and they couldn't solve ok yes so this is true also here so we are also talking about penetration spectra ok, you will see which on what on the Teckmuller curve absolutely, so both motivation the diophantine motivation also generalizes it's a diophantine problem for interval exchange transformations analogous of rotations ok, that's another thing that I like to do but it's also the geometric motivation is penetration spectra generalize that's why we studied this Lagrange spectra ok, so the first key picture if you have a Vich surface in the Vich group you want to look at the space of a fine deformation of your Vich surface so you might have heard from Ronan from last week, you can act on a translation surface by SL2R by applying the matrix to each polygon and gluing after applying linear matrix parallel sites are still parallel you can still glue them so there is an SL2R action and I want to look at the space of a fine deformation and I'm gonna define it like this I'm gonna think of marked a fine deformations as triples of from a fixed S0 to S and I'm gonna say the two triples, two marked triples are equivalent if I can find a translation equivalence which makes this diagram commute something is wrong, ok, it doesn't matter ok, so the space of marked a fine deformation is actually isomorphic to SL2 and SL2 is like the unit tangent bundle of the unit disk picture yourself this disk the unit tangent bundle of this disk as all possible a fine deformation marked a fine deformations of surface, you can actually place the standard triple your surface you can think of it as the center of the disk and as you apply a matrix in SL2R you move on the disk and the which group acts on a marked a fine deformation by changing the marking and it acts actually on the disk by maybe those transformations and you can take the quotient of this disk by the action of the which group and this is an example of a tassillation of the disk where the fundamental domain is the which group and the quotient is the space of a fine deformations and this quotient will be the unit tangent bundle of a hyperbolic surface with casps and this hyperbolic surface with casps is the type molar curve associated to your which surface so I want to think of type molar curves as space of a fine deformations of a which surface and remark this is not the case if I take a random translation surface then the space of a fine deformation will be much larger it will be the whole the closure will be the closure will be typically the whole modular space but this is true for which ok so this picture will be useful and I want to do some kind of use the action of the which group on this disk to understand my dynamical properties so I want to explain some ideas from how you characterize cutting sequences first so the first three slides will not have the disk but then I will show you the disk how it enters into play and I'm gonna do a simplification as I said the result I want to talk about is bone molar classification but I want to tell you first some ideas from the octagon case from our work with Miley which are easier and then I will tell you the difference and novelty in the bone molar case ok again you can start listening from here if you are lost for now so if I give you this octagon so I'll talk about the octagon and then explain the difference with bone molar so let's go back to the regular octagon and to cutting sequences on the regular octagon so first using isometries of the regular octagon I can assume that the linear trajectory I want to study has slope in the sector 0 pi over 8 so this is this blue sector I can always apply an isometry and bring it back there so first I want to do an elementary observation that there are clear restrictions of what a cutting sequence can do so I want to look at possible pairs of letters or transitions so which pairs of letters can follow each other given that my trajectory has a slope close to horizontal for example if my trajectory hits the side A and has slope in this sector it's bound to hit D after A so I cannot go anywhere else than D so from A I can only see D but if I hit this from the side D and it depends on how I come and what my slope is I could either hit A or I could hit B so from D I can go to A or to B those are both allowed transitions and similarly you can record which other restrictions there are and you get a little diagram which I call transition diagram and by elementary geometric restriction in this sequence it has to give me an infinite path on this diagram because it's forced to of course there's far from being a characterization but it's a first necessary condition and the idea is that renormalization allows you to multiply this condition on different scales so first you have to do your exercise if you have other sectors you can get just permutations of the same diagram so these are all diagrams one to the eight possible sectors the other half is symmetric so I don't need to care about and if I give myself a cutting sequence just by looking on which so it will give me an infinite path on at least one at most two of these diagrams so you can kind of recognize the sector already by just this elementary observation then there is a non-trivial operation so I want to say a combinatorial operation which is called derivation and derivation will be an anti substitution so I remember substitution grows the lengths of words derivation will shrink the lengths of words and we will find substitutions as antiderivations so if I have a cutting sequence on the octagon I can say the derived sequence consists of the sequence of sandwich letters what are sandwich letters so sandwich letters are letters preceded and followed by the same letter so here A is sandwiched between D's B is sandwiched between C's and I can run through my sequence and search for sandwich letters and keep only those so the derived sequence consists of only the sandwich letters in my sequence which reduces lengths of finite blocks and so in Smiley we show that the derived sequence is again a cutting sequence if I start with a cutting sequence the derived sequence is a cutting sequence so it has to be admissible on one of the diagrams and conversely substitutions are like anti derivation so if I know the arrival sector so if I know in which diagram the derived sequence is leaves then I can reconstruct my original sequence by a substitution so there are 8 substitutions so that if the derivative leaves in the I's transition diagram the original sequence is substitution I of the derived sequence so derivation is not uniquely invertible but if I know where I land I can invert it through a substitution ok, that's the key step that I want to bring for word for on goes and they're not really substitution they're actually substitution on transitions so you can think of them as you have to change alphabet if you want to really call them substitution and in our first paper we don't talk about substitution you can think of them as morphism of the transition diagram so an arrow is mapped to a sequence of arrows so this combinatorial operation of derivation is nothing else than the symbolic effect of an affine automorphism actually of the affine automorphism that we saw before the stretch is the octagon so this is a geometric meaning of derivation remember this matrix that we saw that sends the octagon in a shear octagon so the claim is that to see that the derived sequences of cutting sequences are cutting sequences you can prove that if I take a cutting sequence in this basic sector the derived sequence is the cutting sequence of the same trajectory with respect to the sides of the shear octagon so the shear octagon will hit my sequence less often and the sandwich letters are exactly the ones which hit the sides of the shear octagon and then if I want to re-normalize if I apply my affine diffiel which sends the shear octagon back to the octagon and then I open my trajectory but I still have the derived sequence is the cutting sequence of a new trajectory on the regular octagon so I can then repeat my procedure now it's time to see the disk picture so this is the disk that molar disk of the octagon so the center is the octagon which is the symmetry of order 8 and this is the fundamental domain the tussillation given by the which group of the octagon and all the red vertices which have order 8 symmetries are shear octagons so they are cut and paste equivalent to the center to the regular octagons so you can think of them as shear octagons and if you choose as generators of the which group I choose generators of isometries times this gamma this shear and this is the telegraph of moves that I want to use so here I can use reflections or rotations in the group and to move along a segment I use my gamma or conjugate of gamma so from the center these are the possible moves I can do so if I want to study a trajectory of a direction on the octagon sorry trajectory on the octagon in direction theta I'm gonna look at the geodesic ray which points to theta in the tekmolar disk and I'm gonna shadow this ray with a sequence of moves from my tree so I'm gonna look at group elements which approximate my geodesic and it will go to a sequence of octagons it will go to a sequence of vertices and I claim that if I derive my sequence what I'm getting are cutting sequences with respect to this stretched octagons so these octagons which approximate my geodesics will be stretched in the direction of my direction and the right sequences are just describing how I hit the stretched octagons and how do I find my substitutions if I can actually label this tree with number is from 0 to 7 and the path that I'm selecting it's telling me which substitutions I have to use to invert derivation so if my labels will be I0, I1, blah blah then my original sequence is the ends derivative with this substitutions backwards so this is how you find your exotic expansion for the octagons it's a triangle group with actually infinity infinity n so it has one elliptic point and two parabolic elements thanks for this question because now what is the difference with Bohm-Mohler Bohm-Mohler has again a triangle group but it has two elliptical elements so SMM has two elliptic points it's a MN infinity triangle group so it has one parabolic point and two points of order M and order N so indeed that's the why it's kind of harder than the square and actually what are the green vertices 3,4 in my picture there are the 3,4 surface which has order 4 symmetry and what are the red vertices in this disk actually turns out that on the same disk there are also affine copies of the 4,3 so on the MN disk there are also the NM so these kind of surfaces are coupled and the red vertices are affine copies of cut and paste copies of 4,3 surface so the key idea is that this time to use the randomization we are not gonna use which group elements because actually Diana tried to describe which group elements and she couldn't get so they are kind of hard to describe so the idea is to break your task into simpler moves that are not made by which group elements but I want to go I want to use this tree of randomization which goes from a green vertex to a red vertex so this is not a which group element will go from green to green so how do I go from green to red well I want to go from MN surface to an MNM surface so by the way sorry this is octagon and two squares and this is hexagon and two triangles and if you shear them a little bit this is octagon and two squares so it's SNM and this is hexagons and two triangles so how do you go from one to another so Hooper proved that there is an affine diffio from one to another so let me show you one cartoon movie you can straighten this picture change diagonals from red to green in this rectangular grid cut and paste some triangles to get the rocket, this is the rocket formation and shear it to get back that so that's how you go from one to another and you see what you have to understand it's a flip of diagonals so this is the last slide that have on not the last, the one before the last so what do we do? we also define derivation combinatorially I'm not gonna tell you what the derivation looks like combinatorially but our derivation goes in two steps I want to go from MN to NM and back so first time I derive I'm gonna send a cutting sequence of a trajectory here to a cutting sequence of a trajectory there so derivation with map cutting sequence is here to cutting sequence is there and it comes from with the recipe for substitutions so if I know where the arrival the sector where the arrival direction is and I can recover this combinatorially from transition diagrams like I did before then I can also invert and write down explicit substitutions and we give recipe to produce substitutions for every MN so I can find substitutions to go back substitutions in quotation marks because technical details so what's the picture on the disk this is the last picture and then I have only two slides on hall so that's the picture of the disk so let me remember green vertices are one surface and red are the other surface this is my tree of wolves I want to study trajectory in direction theta I'm going to look at a ray geodesic ray going to theta and I'm going to select a pass in my tree which approximate my geodesic this will give me the normalization moves I have to do and again like before I can interpret these combinatorial derivatives as cutting sequences of the same trajectory to this shared polygonal presentations and here the polygonal presentations for M and odd for k odd they will be the formation of one surface for k even they will be the formation of the other surface I always jump from one type to another type and again you can label this graph explain how to do that and use the labeling of this graph to read off which substitutions you want to use and if you like you can interpret the action of these three of wolves on the boundary as a continued fraction on directions and these substitutions you need to use are governed by the symbolic by the entries of this generalized continued fraction for bomb holder yes actually this is really I'm taking my translation surface I have a direction and I'm just looking at the geodesic flow which shrinks that direction so I'm rotating myself in that direction and that if I start from the whichever convention you pick I start from the center of the octagon and you're gonna give me a geodesic ray so in reality with this convention you know theta goes from 0 to pi you do a full angle so this will be like if you pick reference point will be like 2 theta or something explicit but it's really I want to pick the geodesic which renormalize my linear trajectory ok so now I have just two slides to finish with some ideas about whole rays so break the last again and there's a last recovery point for the last idea so again here we want to work with this like molar disk of renormalization moves and we want to show I will remind you what the Lagrange spectrum is because it will have forgotten so I take any which surface here so I don't know the which group I can take a fundamental domain for the which group but we saw before that it was very nice to have ideal tussillations so what I'm gonna do if there are elliptic points I'm gonna look unfold the elliptic points and look at the super tussillation which is ideal so I notice I don't want to pass to a finite cover of my which surface which I could I want to study my original surface but I'm gonna use this super tussillation to call to the geodesic flow so my picture has an ideal tussillation which could be a finite cover of my fundamental domain and this ideal fundamental domain there are gluings given by elements of the which group and if I want to study the Lagrange value on a certain direction again I'm gonna pick my geodesic ray which shrinks that direction and I want to call the geodesic ray by the sequence of sites of my tussillation hits or by the sequence of which group elements which glue the sites hit and because this is an ideal tussillation equivalently I can look at my endpoint of my geodesic and there will be a sequence of arcs boundary arcs boundary arcs here here nested boundary arcs which contain this point so out of the sites of my tussillation should imagine this fundamental domain reflected and I'm gonna pick the arcs which cross nested arcs which crossed my ray and shrink to that point so this boundary arcs is what I'm gonna use and if you know Bowen and series boundary expansions this is actually the easy case of Bowen series because I'm looking at an ideal tussillation if you know Bowen series ok, so if I have a sequence of boundary arcs which come from this ray I'm gonna associate to boundary arcs pairs of saddle connections on my surface like this which contain the direction theta so if I have a boundary arc it has two endpoints, two angles and I can look at actually turns out that this vertices of my ideal tussillations correspond to directions where I have saddle connections on my surface so in those directions which are endpoints I have saddle connections and I'm gonna pick the shortest saddle connection in that direction this will give me a wedge a pair of saddle connections which contains the vertical and as I go to a nested arc as I go to a nested arc a shrink wedge so I will get a sequence of saddle connections which converge to the vertical direction ok, it's a sequence given to me by renormalization by the Vitch group and it's a sub sequence of all saddle connections which approximate theta but I claim that this saddle connection see the spectrum they see the top of the spectrum so remember the spectrum was the sup of one over the area so saddle connections as they approximate theta so so the key lemma is that there exists for high values in the spectrum after a certain L0 I can just use these wedges I can take a direction theta and the sequence of wedges that approximate it and only computing this area of these two saddle connections right and left and the limit as n grows I am missing some saddle connections but those are the important ones that allows me to compute to the spectrum and this is kind of the part I wanted to tell you this is my last slide from here on it goes back to kind of redoing whole argument in a new setup so whole uses a lot that there is a formula using continual fraction for the spectrum so if you know the continual fraction entries so this spectrum is the lim sup of this formula so I have an, here is an integer and then you have an infinite continual fraction of future entries and the finite backward continual fraction of this parentheses are continual fraction so what we do we define as continual fraction we define continual fraction in our setup and this s stays for a modified continual fraction and we show that these areas can also be computed by a similar formula in terms of continual fractions and then whole has a proof which uses arguments on sum of counter sets you can prove that for bounded entries this tail corresponds to counter set of values these other tails correspond to another counter set but if the counter sets are have large halves of dimension the sum of two counter sets contains an interval so this is kind of redoing the counter set proofs and I have a hidden half slide which explains more but I think you are all tired and this is a good point to stop and I showed you the geometric argument using the tekmuler disks so I hope I convince you that tekmuler disks and which groups are useful to study dynamics on which surfaces thank you