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e a famous article asking whether can one hear the shape of drum mathematically speaking one wants to reconstruct up to isometries a domain from the knowledge of the spectrum of its laplacian in his article m kac showed that one can hear the shape of a disk because of the following two facts the area and the perimeter of a smooth domain are determined by the eigenvalues of its laplacian via the asymptotics of the trace of the heat operator the isoperimetric inequality says that we can recognise the disk from its perimeter and area indeed any smooth domain satisfies and the equality holds if and only if is isometric to a disk of radius in particular a smooth domain with the same laplace eigenvalues of has area and perimeter so that the isoperimetric inequality ensures that and are isometric in a preprint from 2019 hezari and zelditch showed that one can also hear the shape of an ellipse of small eccentricity as it is explained by zelditch in this video here a first order approximation to their basic strategy to hear ellipses of small eccentricities is to replace trace of the heat operator and isoperimetric inequality in kac s argument by trace of the wave operator and dynamical rigidity of ellipses more concretely hezari and zelditch considered a smooth domain which is isospectral to an ellipse i e for all of small eccentricity and they took the following steps in section 2 of their paper it is shown that is necessarily close to in fact after parametrising with an arc length parameter we can control the norms of the derivatives of the curvature of from the asymptotics of the trace of the heat operator indeed where and in general for a certain constant and an adequate universal polynomial here is the th derivative of with respect to in particular since and are isospectral for all and melrose explored this fact to get a pre compactness bound for all via a bootstrap argument where the poincaré inequality and the sobolev embedding theorem are employed to convert bounds on into bounds on in other terms after melrose the shape of any isospectral to is bounded by reworking melrose s argument hezari and zelditch actually show that is almost circular in the sense that for all in view of a theorem of avila de simoi and kaloshin an almost circular domain is isometric to an ellipse provided is rationally integrable i e for each the periodic trajectories with rotation number in the billiard table determined by are tangent to a smooth convex curve usually called caustic in sections 3 to 6 of their paper it is shown that the portion between and of the singular support of the trace of the wave operator coincides with the set of lengths of periodic trajectories with rotation number in an almost circular billiard table in particular if is isospectral to one concludes in section 7 of their paper that for each all periodic trajectories in with rotation number have the same length and they form a caustic so that is rationally integrable and a fortiori isometric to after avila de simoi kaloshin a natural question raised by hezari zelditch work is to determine the magnitude of the upper bound on the eccentricities of the ellipses which can be heard from their methods in this direction i would like to conclude this short post by noticing that i asked a group of 6 undergraduate students in their 2nd year at ecole polytechnique to follow closely the articles by avila de simoi kaloshin and hezari zelditch while trying to explicitly compute as many implied constants as possible and after 6 months of work they produced this report here in french concluding that is not bigger than of course there is plenty of room for tiny improvements here but one will probably need some new ideas before reaching a normal size e g leave a comment posted in uncategorized posted by matheuscmss april 22 2020 bistro seminar simion filip curtis mcmullen martin moeller and i are co organizing an online seminar called bi lliards s urfaces à la t eichmueller and r iemann o nline bistro similarly to several other current online seminars the idea of bistro arose after several conferences meetings etc in teichmueller dynamics were cancelled due to the covid 19 crisis in any case the first talk of bistro seminar will be delivered tomorrow by kasra rafi at 18h cest in a nutshell he will speak extend the scope of a result of furstenberg on stationary measures previously discussed in this blog here and here to the context of mapping class groups closing this short post let me point out that the reader wishing to attend bistro seminar can find the relevant informations at the bottom of the seminar s official webpage leave a comment posted in conferences tags bistro seminar posted by matheuscmss april 16 2020 maximal entropy measures and birkhoff normal forms of certain sinai billiards sinai billiards are a fascinating class of dynamical systems in fact despite their simple definition in terms of a dynamical billiard on a table given by a square or a two torus with a certain number of dispersing obstacles they present some nasty features related to the existence of grazing collisions placing them slightly beyond the standard theory of smooth uniformly hyperbolic systems the seminal works by several authors including sinai bunimovich chernov paved the way to establish many properties of sinai billiards including the so called fast decay of correlations for the liouville measure a feature which is illustrated in this numerical simulation due to dyatlov here nevertheless some ergodic theoretical aspects of certain sinai billiards were elucidated only very recently for instance the existence of an unique probability measure of maximal entropy was proved by baladi demers in this article here an interesting remark made by baladi demers is the fact that the maximal entropy measure should be typically different from the liouville measure in fact forces a rigidity property for periodic orbits namely the lyapunov exponents of all periodic orbits coincide note that this is a huge number of conditions because the number of periodic orbits of period grows exponentially with and in particular there are no known examples of sinai billiards where in a recent paper de simoi leguil vinhage yang showed that forces another rigidity property for periodic orbits namely the birkhoff normal forms see also section 6 of chapter 6 of hasselblatt katok book and appendix a of moreira yoccoz article at periodic orbits whose invariant manifolds produce homoclinic orbits are all linear in particular they proposed in remark 5 6 of their preprint to prove that for certain sinai billiards by computing the taylor expansion of the derivative of the billiard map at a periodic orbit of period two and checking that its quadratic part is non degenerate in this short post we explain that the strategy from the previous paragraph can be implemented to verify the non linearity of the birkhoff normal form at a periodic orbit of period two of the sinai billiards in triangular lattices considered by baladi demers remark 1 in our subsequent discussion we shall assume some familiarity with the basic aspects of billiards maps described in the classical book of chernov markarian 1 sinai billiards in triangular lattices we consider the sinai billiard table on the hexagonal torus obtained from the following picture extracted from baladi demers article here the obstacles are disks of radii centered at the points of a triangular lattice and the distance between adjacent scatterers is as it is discussed in baladi demers paper the billiard map has a probability measure of maximal entropy whenever where the limit case corresponds to touching obstacles and the limit case produces infinite horizon 2 billiard map near a periodic orbit of period two we want to study the billiard map near the periodic orbit of period two given by the horizontal segment between and a trajectory leaving the point in the direction travels along the line until it hits the boundary of the disk of radius and center for the first time by definition the time is the smallest solution of i e hence where and the billiard trajectory starting at in the direction hits the obstacle at where and after a specular reflection it takes the direction as it is explained in page 35 of chernov markarian s book this geometric description of the billiard map near the periodic orbit of period two allows to compute along the following lines if we adopt the convention in chernov markarian s book of using arc length parametrization on the obstacle and describing the angle with the normal direction pointing towards the interior of the table using a sign determined by the orientation of the obstacles so that this normal vector stays at the left of the tangent vectors to the obstacles then the billiard map becomes where and for and moreover and where since we also have and it follows that where and in particular the determinant of the quadratic part of the taylor expansion of near the periodic orbit of period two is thus for one has so that the quadratic part of the taylor expansion of is non degenerate because for remark 2 in the limit case leave a comment posted in math ds mathematics tags birkhoff normal forms maximal entropy measure sinai billiards posted by matheuscmss february 3 2020 decrease of fourier coefficients of stationary measures on the circle after jialun li last january 8 2020 jialun li gave the talk decrease of fourier coefficients of stationary measures on the circle in the flat seminar that i co organize with anton zorich once per month in this post i ll transcript my notes of this nice talk while taking full responsibility for any errors mistakes in what follows 1 introduction 1 1 stationary measures consider the linear action of on induces an action on the projective space for later use recall that via given a probability measure on we can build a markov chain random walk whose steps consist into taking points into where is chosen accordingly with the law of the absence of hypothesis on might lead to uninteresting random walks in fact if a point is stabilized by two elements then the random walk starting at associated to is not very interesting for this reason we shall assume that hypothesis i the support of generates a zariski dense semigroup remark 1 by tits alternative in our current setting of the hypothesis i can be reformulated by replacing zariski dense with not solvable as it was famously established by furstenberg the random walks associated to have a well defined asymptotic behaviour whenever i is fulfilled theorem 1 furstenberg under i there exists an unique probability measure on such that for all as here the convolution of with a probability measure on is a probability measure on defined as so that is the distribution of points obtained from after steps of the markov chain associated to in the literature is called furstenberg measure and it is an important example of stationary measure i e a probability measure on which is invariant on average 1 2 lyapunov exponents the stationary measure can be used to describe the growth of the norms of random products associated to almost every whenever satisfies i and its first moment is finite theorem 2 furstenberg guivarch raugi if has finite first moment i e and satisfies i then for almost every the quantity is called lyapunov exponent 1 3 regularity of stationary measures the furstenberg measure dictates the distribution of the markov chains associated to and for this reason it is natural to inquiry about the regularity properties of stationary measures in this direction guivarch showed that the furstenberg measures have a certain regularity when satisfies i and its exponential moment is finite hypothesis ii there exists with theorem 3 guivarch under i and ii there are and such that for all and where is the interval of radius centered at in particular has no atoms more recently jialun li established in this article here another regularity result by showing the decay of the fourier coefficients where more concretely he proved that theorem 4 li under i and ii we have in other words is a rajchman measure in a certain sense the role of assumption i in the previous theorem is to avoid the following kind of example example 1 let with note that the semigroup generated by is not zariski dense in as and are upper triangular we affirm that there is no decay of fourier coefficients in this situation indeed recall that if we identify with via then acts on via möbius transformations i e an element acts on as in particular and the fourier coefficients of the stationary measure given by the standard hausdorff measure on middle third cantor set do not decay to zero in a similar vein if is a real number such that is a pisot number then with admits a stationary measure called the bernoulli convolution of parameter describing the distribution of the points where with probability whose fourier coefficients do not decay the proof of theorem 4 is based on a renewal theorem more concretely given a function let by thinking of as a smooth version of the characteristic function of an interval we see that is counting random products with norm in the interval in this context guivarch and le page established the following renewal theorem theorem 5 guivarch le page under i and ii one has as remark 2 another important fact in the proof of theorem 4 is the non arithmeticity of the jordan projections of the elements of i e the fact that these jordan projections generate a dense subgroup of whenever i is satisfied since we will come back later to the discussion of deriving the decay of fourier coefficients e g theorem 4 from a renewal theorem let us now move forward in order to introduce the main result of this post namely a quantitative version of theorem 4 2 quantitative decay of fourier coefficients the central result of this post is inspired by the following theorem of bourgain and dyatlov theorem 6 bourgain dyatlov if is the patterson sullivan measure associated to a schottky subgroup of then there exists depending only on the dimension of i e the hausdorff dimension of the limit set of the schottky subgroup such that for all the method of proof of this result is based on the so called discretized sum product estimates from additive combinatorics interestingly enough this result can be interpreted as a decay of fourier coefficients of certain stationary measures thanks to the following theorem theorem 7 furstenberg sullivan the patterson sullivan measure of a schottky subgroup coincides with the stationary measure of some probability measure on satisfying i and ii remark 3 we saw the proof of a version of this result for cocompact lattices of in proposition 14 of this blog post here the previous theorems suggest that a decay of fourier coefficients of the furstenberg measure associated to a probability measure on satisfying i and ii this statement was recently proved by jialun li in this article here theorem 8 li if is a probability measu...