 Okay, so thank you for that. I'm very happy to be here. I want to thank Stefano for the invitation to talk at this conference. I also want to have the opportunity to say how happy I am to be here. I mean, I think this is one of the most fantastic institutions we have in Italy. I mean, it's not an Italian institution. It's very much supported by the Italian government. But I think really this is an extraordinary story. The ACTP is very nice. It does a fantastic job. I myself owe a lot of my knowledge and my career in my networking to having been a student, like some of the younger people are here 27 years ago. So I'm really happy to say thank you, ACTP, and keep on doing this fantastic job. Now, after this short introduction and celebration well-deserved of ACTP, let me tell you a little bit what I want to do. So actually what I want to do is to provide a little bit that the title is frightening and not is very inspiring. So I'm grateful to all of you who despite the sun are here. What I will try to do is to put this a little bit in perspective. So tell you why after, I mean, since 10, 15 years there is a certain number of people who are trying to do certain generalizations of results which have been painfully obtained in the case of circle maps to a generalization of circle maps which are interval exchange maps. Okay, so let me start with a short introduction. Just a little bit. Okay, so I will fix a little bit of notation on the oftentimes numbers and continued fractions because I will not need them a lot but I will lead them a little bit in the future. So if alpha is in a rational number, we all know that we can associate to alpha its infinite continued fraction. This is actually unique and it's this writing of where all the A-note is a relative integer and all the other AIs are actually integers which are actually positive. And so there is actually a unique way to write this and we denote normally these AIs are called the partial quotients and then if you truncate at order n you're paying what are called the convergence and there is a characterization of the oftentimes numbers so let me just recall you that if tau is negative the set of the oftentimes numbers with exponent tau is actually the set of irrational numbers such that there exists a positive constant gamma for which one has alpha minus p over q greater or equal to gamma over q2 plus tau for all rational p over q. I will try. And this is the same as the set of irrational numbers so you can just use the growth of the denominators of the convergence which always grow and condition exponentially fast as a characterization saying that it's the same as qn plus 1 b go of qn to tau. Okay, as soon as tau is positive then this has a full measure. It's a full measure set in the real line. Yes, thank you. One plus tau, absolutely. Equivalently you can write a m plus 1 b go of qn to the now with tau. Thank you very much. And there is a class which is to some extent if tau is zero then the set is called constant type and it has not full measure and there is a class which is sort of intermediate of this growth type which is just the intersection for all positive tau of this eta. So basically a number is of growth type. If for all positive tau there is a positive constant gamma such that this is true for all p over q. So growth type is called this way because there is a celebrated theorem by Roth in 1955 which is the end of a long story. But it's still too small. That irrational algebraic numbers belong to Rc. Okay, so you may wonder what does this have to do with dynamics but it has to do with dynamics. For the following reason let's consider now circle maps. So just let's start from the simplest case which is the rotation of the circle which is also just to start to talk about interval exchange maps. If you have an interval exchange map if you forget about periodicity because if you look at the graph of a rotation of an angle alpha so fundamentally if this is the diagonal what you get is that if you forget about periodicity it splits into parts. This is 1 minus alpha and if you call a and b these two intervals you can see the action of the mass as per muting these two intervals is the only way you have to exchange intervals if you have only two. So this is just to remind that the simplest case of interval exchange map is a rotation. Now if you have the rotations you can actually characterize the set of Roth numbers as well also as the set of the oftentimes numbers I will just call dc the union instead of the intersection of dc tau. So there are two easy theorems that you can prove. I mean the first theorem this is really easy if you want an exercise is that for all an alpha being the oftentimes is equivalent to the fact that for all function phi it is infinity over the circle and of the aromene there exists a unique function psi which is also infinity on the circle and of the aromene such that psi composed with the alpha minus psi is equal to phi. This is easy. This is not so easy at all. It's a theorem actually which I will call theorem one which is due to fundamentally I think Herman, Rusman and I think also Jocos but he denies it but I think it's also Jocos which says that you can equivalently characterize Roth numbers in a similar way namely alpha is a Roth number is a Roth type the statement is a little bit less sexy I mean but it's still reasonably elementary for all rs real numbers such that r is greater or equal sorry is greater than s plus 1 equal to 1 and for all function phi which is of class cr over the circle and of the aromene there exists a unique psi in cs which solves the common this will be called this is the homological equation I will say more about this in a second such that which is also and of the aromene such that this is true okay just to tell you in a nutshell what is the idea behind the proof of both theorems the idea on the circle is very elementary then it's also elementary to get there it's not so elementary it requires more sophistication to get there but it's not terribly complicated it is just that you use Fourier analysis on the circle very easily and you rewrite this equation at the level of the Fourier coefficients and you see that what happens is that psi hat of n times e to the 2pi i and alpha minus 1 has to be equal to phi hat of n for all n integer non-zero and now well now this behaves fundamentally how well if you know that alpha is the often time of exponent tau then this essentially this will lead you to the estimate that psi hat of n will be bounded by some constant times phi hat of n times absolute value of n 1 plus tau okay and then if you at least one implication this arrow here is going to be clear to you because being sinfinity means that the Fourier coefficients decay faster than any power so I mean one arrow is clear if you want to get the other arrow you have to build counter examples but that is not very complicated on the other hand if you want to work as you do here with holder differentiability then you have to do a substantial work using Pallet-Litterwood theory under my interpolation to prove this statement so this is really a theorem I'm going to feed a theorem it's not just an exercise but you can do it what is also clear is that if you are interested just to the sobola kind of estimates then this is all what you need because sobola spaces for periodic functions are really usually characterized by looking at Fourier coefficients and the way they behave so in sobola scale this is transparent okay so one of the goals I will have to take motivation for the title of my talk is to provide not at all unfortunately an equivalent statement for all general for all interval exchange maps but to provide a partial result in this direction okay a result which goes in the direction of getting here it's weaker it contains a weaker statement of this in the circle but it is in the same spirit now why is this interesting what's the motivation behind looking at the chronological equation in case you need one which is after all legitimate I do like it so I don't need it very much but I'm probably an old guy most regular guys will say why the hell are you doing this okay the reason for which you are interested to the chronological equation is rather clean and it has to do with the following thing it has to do with the fact that after all if you think of the circle the role of the rotation is so sound so important that you may ask yourself the question of what is what dynamical systems are actually are indeed conjugate to rotations so what you really would like to understand is the conjugacy classes of the rotation and this is actually in the case of the circle one of the very rare cases where we do have a lot of information we have a full characterization in this infinity in the analytic case and there is also a lot of information in the holder case which is very rare you could be less ambitious to begin with and instead of asking for the full conjugacy class of the rotation you could ask at least what happens locally so you may say okay let me now try to move from the rotation to something more general let me take F which is an orientation preserving the field of class C R of T let me assume it that C R close to our alpha and let me ask, I mean let me if you want denote with C R S lock so lock refers to this characterization the set of the F like this such that there is this H which is a C S B fill of the circle such that conjugates F to our alpha if you drop the assumption of being C R close to the rotation then you just will have C R S and then fundamentally there is a second theorem which has many names at least from my knowledge I apologize some of the names are in the room if the attribution is fuzzy but fundamentally it's a theorem which owes to Herma Katz-Nelson Ornstein Kahnien Sinai Jokot and maybe someone is missing but I apologize we say the following try to say it cleanly DC R minus S minus 1 minus epsilon is contained in CRS is contained in CRS lock is it large enough can you read something I see people nodding at least in the first four lines of fire I owe you a beer ok thank you I keep going someone is also nodding in the top DC R minus S minus 1 plus S so basically tells you so this is actually what is the global conjugacy theorem this is the local class of course but it I mean it's major achievement that also in the holder class you can characterize not exactly these sets for example we don't know if these two are different or the same we don't have equalities we don't know exactly what are these classes but at least we have a good amount of information so what I will also try to tell you is the statement of the theorem it's a space thank you it's a space of alpha such that for all F it's a space of rotation numbers you want to characterize rotation numbers thank you for all F like here thank you very much ok in particular just to give you a little flavor of how Roth type gets into the thing what happens is that Roth type is included in C lock yes yes yes yes yes yes and the rotation number of F is alpha yes thank you absolutely otherwise it's a strange thing r r minus 1 minus epsilon for all epsilon positive ok so this is basically if you want a special case of 2 and the goal of the next 25 minutes will be to provide you also the statement of an analog of theorem 1 and the statement of an analog of theorem 2 neither neither the analog of theorem 1 or theorem 2 are fully satisfactory but at least go beyond the circle case ok let me just remind you that the two questions I mentioned namely are these sets are these two classes the same and to and to say exactly what they are at least in the local case or in the global I mean just to characterize exactly the set of rotation numbers which for which you have this are open and are explicitly stated in the proceedings of the of a school on small devices we held in Chitravi in 1998 so I just want to draw the attention of young people the quasi periodic dynamics even in the circle case there is still a number of questions open it's not a closed field it's not over at all to be completely honest I don't know I mean I think that we don't in general know if they are different or not I mean the full understanding of the thing is not there I don't master all the literature to be honest so I'm not 100% sure that I may miss something which has been happening but I don't think I think I have good faith in the fact that the saying that the problem is still open is a fair statement it seems but it seems it's infinity sure okay it's infinity but no they don't because there are two different conditions I mean Bruno and condition H okay now very quickly let me get to interval exchange maps now so we move from circle maps to interval exchange maps so I mentioned that when you have a circle map you do have an interval exchange map that it's a map which exchanges two intervals as I tried to convince you with the drawing I'm leaving on the blackboard but now let me give you a proper definition so T so you have an interval so you have a map of an interval into itself and you will say that it is an interval exchange map if the following is true well it's simply one to one and locally a translation except at finitely many discontinuities I also want to define taking advantage of colors a generalized generalized interval exchange map as a map of an interval which is one to one and locally I replace a translation with a orientation prefers orientation reserving homeo so basically for example in this case it would be instead of doing this you do something like okay so to each interval exchange map you can associate a few numbers and basically two data which completely characterize the parameters you have you need to specify them so the two numbers are first of all D which is just the number of intervals so this is the case D equal to 2 but you may have exchanging 3 or 4 more intervals of course and then there are other numbers you want to associate which are called the genus G and the number of market points S which are somehow as the name suggests related to the fact that interval exchange maps have the same relationship to the flow and translation surfaces which have been described by Anton Doric in his talk last week and which certainly will reappear I think this week in other talks by Corinna or Sasha I don't know but I really would like to be minimal on that because I am already short in time but anyhow the idea is that the first thing to say is that in the exchange map you associate some combinatorial data because in the case of two intervals there is not much combinatories to say you have two intervals and the only way you have to exchange them is to permute them but of course when you have 3 or more intervals you have several ways of permuting them so fundamentally what you do is that you label the intervals with some alphabet A whose cardinality is going to be D I will denote them either T or B for top and bottom which tells you how the intervals labeled by the letters in A which typically are A B C D E F say that D is equal to 6 are put one after the other before and after the application of the map ok so for example in this case you would have A B and B A and the two maps would be the first one sending A into 1 and B into 2 and in this case sending B into 1 and A into 2 because you are exchanging the order and now you have two bijections from A to 1 D which tell you what are the order of the intervals before and after you applied the map and then the other data you need apart from the combinatorics are the length data so actually you have an interval which will be split in this way and you will have basically lambda alpha which is going to be normally denoted the interval I T A I T B and I T C T refers to top line and bottom line and what happens is that the map sends the points in this way of course lambda alpha is the length of I alpha T which is going to be the length of I alpha bottom for a standard because you are just doing it by translation so the length data which is a vector of positive real numbers of D positive real numbers and the two permutations completely characterize the parameter space now as I mentioned D is the number of intervals S is the number of market points in the invariant in the translation surface you can build by suspending the interval exchange map in a similar way that you can build a translation surface for starting from a rotation and G is the genus and there is a relationship between these three numbers G actually only depends and also has on the permutation and the relation being that G is equal to D minus S plus 1 over 2 so for example this is still an example this A B C going into C B A is still an example in genus 1 with two market points D is equal to 3 if you want however you can build examples in any finite genus if you make A B C D going into D C B A you get D equal to 4 genus equal to 2 and that is equal to 1 if you make A B C D E F into let me make it less standard than usual A F C B you now have D equal to 6 S equal to 3 and therefore the genus is still equal to 2 but you can play around with this in many many ways okay let me now give you I would like not to erase them if I manage the statements would generalize theorem 1 and theorem 2 in later you would say I will finish it with a little bit of our statement because they are not as precise as theorem 1 and theorem 2 but you try to go beyond the circle and torus case to the general case of interval exchange maps and of arbitrary surfaces so I start actually from the statement which tries to generalize theorem 2 with this statement and this is actually permussin Jean-Christophe Yokoz and myself it appeared in 2012 so it's a if you want it's a local linearization theorem for generalized interval exchange maps okay so you start with a standard interval exchange map T0 which we call of restricted rough type we will hopefully say something about what does it mean it includes in D equal to 2 the case of rough type so it boils down to rough type when you are on the circle but you have to make several other assumptions when you want to deal with general interval exchange maps fundamentally what happens is that you have to build some sort of generalized continued fraction which you will use to understand the orbits the proof will be fundamentally to try to solve I mean to prove using a certain trick the homological equation the trick is due to Hermann is called the Schwarz and derivative trick but there is a way somehow to reduce to a sort of fixed point theorem this thing using the homological equation and you need to build the solution to build the solution you have to understand how certain orbits give rise to Birkhoff sums eventually this is done by using matrices which generalize the matrices you have from the continued fraction which are not 2 by 2 positive matrices anymore but they are larger as in they are larger they have more complicated spectrum on which you need to put some assumptions otherwise you will not get out of it because if you have just a nestle to that matrix fundamentally it will start growing or you will have one positive it will be hyperbolic naturally and you will be happy but in this case you have to be more careful and this is what this restricted type copes with I mean it includes the standard case of the circle but it is a little bit more sophisticated ok now take R to be an integer at least equal to 2 and let T be a CR plus 3 simple deformation of T0 ok what does this simple deformation mean well basically it means what you see on the blackboard in the 2 case T which is the red thing coincides with T0 near the discontinuity points actually a little bit better than this you really want them to coincide in the neighborhood of the discontinuity points ok this is not really needed but I just want to give the statement in this case so think of T sensibly if you want as a simple deformation refers to this this assumption of simplicity could be avoided you can get there with some derivative but then you have an invariant which has to be preserved I don't want to talk about that let me just mention this it's a sort of bona fide simple perturbation of the linear case ok just just with no strings attached so this theorem says essentially that the set of the simple deformations which are CR conjugate so unfortunately you use 3 derivatives to T0 is a C1 sub-manifold by a CR close to identity conjugation is a CR is a C1 sub-manifold of co-dimension and the co-dimension is not too bad but certainly bigger than what you have in the circle case is G-1 times 2R plus 1 just notice that this co-dimension is fine if G is equal to 1 because this vanishes ok and you are just left with S so in the circle S would be 1 and you would be happy because the co-dimension in the circle case is simply that you need to fix the rotation number and to have the perturbation of zero mean ok so this is exactly what you're seeing the map is close to yes this is a this is also otherwise you will never be able to conjugate by something which is CR close ok absolutely well I'm describing if you want this C1 is a set of T which are CR plus 3 simple deformations and so on and so on so the set of the TCR simple deformations which are CR conjugate ok is it clear now one motivation for what we have been doing and for the statement of what generalizes theorem 1 so what I would call theorem 1, 2 yes yes well or you have higher co-dimension if you are dealing with interval exchange maps with more than three intervals and you have more market points ok the idea the idea is that essentially somehow the market points generate also co-dimension they also generate what are called for the invariant distributions which become more explicit in what we do I'm putting for the moment I hope I will be able to make a comment in the last two minutes a huge amount of work without which this would be inconceivable ok this doesn't come out of the blue but it is possible because there has been amazing progress in understanding the dynamics of the Tajmila flow and of translation surfaces and renormalization due to the work of many people starting from the pioneering work of Mazer and Beach and then Konsevich, Zorich, Giovanni Forni actually has been doing the main contributions in making small divisor theory enter the theory of translation surfaces so I'm neglecting a lot of work for the time being because I was trying to give a guide lecture of the two statements I wanted to give you but I will comment on this later hopefully ok there is clearly an open problem which motivates a little bit what I'm about to state which is a weaker generalization of that one so why is this the analog of this well because this Roth type generalizes to this condition which is here the setting is of interval exchange maps and of generalized interval exchange maps which generalizes respectively rotations and defils of the circle close to rotations which are inside this and what we are stating is basically a theorem which is similar to this one except that we have CR CR-3 ok which is definitely much worse than what is here and which leads us exactly to the open problem which would be to match this statement namely can one prove it problem, question is it true for CR plus 1 plus epsilon simple deformations ok because this would make really the parallel of this we don't know we don't know but at the level of the homological equation we are somehow getting a little bit closer to this not quite but a little bit in the sense that we are getting a little bit closer but not exactly to one of the two arrows here there are two arrows this is very beautiful at least to certain eyes I find it and a few others in the world find it very beautiful because you have a purely mathematical thing which is characterized in terms of a solution of a dynamical equation so in general it tells you that if you look at problems in a certain way dynamical systems or certain equations which are natural in dynamical system theory provide you equivalences to things which are natural in number theory which is sort of bridge which we maybe don't understand till the end but it's always very nice to have it so what I will now state is a weak generalization of the theorem to integral exchange maps where we basically get a generalization certainly not of the if and no leave but of one of the two arrows unfortunately not quite as beautiful as this one so let me give you the second statement well I think I have to erase this so I won't do it okay so this theorem is actually the one which gives the title to today's talk and it's a theorem that Jean-Christophe and I proved last year okay and it's the following precise statement of the homological equation so let T0 be an integral exchange map of as I mentioned before restricted rough type okay and then for all are greater than greater than one there exists a positive delta and the subspace of CR of okay let me write it in this way and tell you immediately what it is okay we have T0 and we denote with UI the breakpoints of the intervals okay so we have the interval I and then we have U0 and UD which are the endpoints then you have U1 U2 up to UD-1 okay and then what I'm saying here is that I'm taking just the product of functions which are CR on each of these close intervals okay and I call it in this way okay the co-dimension of F is going to be G plus S minus plus number of market points and then there is a linear operator and there exists a linear operator L which goes from F to hold their continuous functions over the whole interval such that if Psi is the image of L of Phi under this then Phi is equal to Psi run T minus Psi for homological equation for all Phi yes this is what I tried to say but I said it very quickly and probably it was not even clear to myself so I have the interval I and I have the I split it into D intervals according to the top there are two partitions before and after application of the map let's focus for a second to the one before the application of the map I have D points UD and UD are the endpoints of the interval which is fixed and from U1 to UD minus 1 you have the other endpoints and with this I denote the space the product of functions it's actually the Cartesian product of the functions which are CR on each of these closed intervals a function defined over there but there are also the restriction to each of the closed intervals is CR they don't need to mention okay on the other hand what you get here is all the continuity everywhere this is what you need so the other continuity here is on the interval itself so it's to give a proper statement but fundamentally it is telling the same thing of this RO with several caveats the caveat is that for all R there exists a delta whereas here you have simply a relation what you are saying here is that basically all what you need is S to be smaller than R minus 1 that's not quite what we get because this delta will depend on R it's not independent so fundamentally it's true that if R is just a little bit above 1 then we get a little bit of older continuity so we are closing spirit to that one but I can't say that this stays true if R is big it may happen that R is 20 but R is still 6 so the gap could grow with differentiability and it's only one error we don't get the other error well there was a just to show you, I'm honest there was a fifth page which was a relation to Forney and open problems but my time is over, I apologize and thanks a lot