upgrading the local ergodic theorem for planar semi dispersing billiards the local ergodic theorem also known as the fundamental theorem gives sufficient conditions under which a phase point has an open neighborhood that belongs mod 0 to one ergodic component this theorem is a key ingredient of many proofs of ergodicity for billiards and more generally for smooth hyperbolic maps with singularities however the proof of that theorem relies upon a delicate assumption chernov sinai ansatz which is difficult to check for some physically relevant models including gases of hard balls here we give a proof of the local ergodic theorem for two dimensional billiards without using the ansatz

quantum properties of irrational triangular billiards triangles with sides given by consecutive integers n n 1 n 2 are fully irrational all angles irrational with pi if 3 120 and eventually the occurrence of gaps in the distribution for n 180 define the onset of a long crossover towards the sequence observed in the integrable limit of the equilateral triangle n infinity

hausdorff dimension of the set of nonergodic directions it is known that nonergodic directions in a rational billiard form a subset of the unit circle with hausdorff dimension at most 1 2 explicit examples realizing the dimension 1 2 are constructed using diophantine numbers and continued fractions a lower estimate on the number of primitive lattice points in certain subsets of the plane is used in the construction

the set of non uniquely ergodic d iets has hausdorff codimension 1 2 we show that the set of not uniquely ergodic d iets with permutation in the rauzy class of the hyperelliptic permutation has hausdorff dimension d frac 3 2 in the d 1 dimension space of d iets for d ge 5 for d 4 this was shown by athreya chaika and for d in 2 3 the set is known to have dimension d 2 this provides lower bounds on the hausdorff dimension of non weakly mixing iets and with input from al saqban et al exceptional directions for the teichmuller geodesic flow and hausdorff dimension 2017 arxiv 1711 10542 identifies the hausdorff dimension of non weakly mixing iets with permutation d d 1 ldots 2 1 when d is odd

non ergodic z periodic billiards and infinite translation surfaces we give a criterion which proves non ergodicity for certain infinite periodic billiards and directional flows on mathbb z periodic translation surfaces our criterion applies in particular to a billiard in an infinite band with periodically spaced vertical barriers and to the ehrenfest wind tree model which is a planar billiard with a mathbb z 2 periodic array of rectangular obstacles we prove that in these two examples both for a full measure set of parameters of the billiard tables and for tables with rational parameters for almost every direction the corresponding directional billiard flow is not ergodic and has uncountably many ergodic components as another application we show that for any recurrent mathbb z cover of a square tiled surface of genus two the directional flow is not ergodic and has no invariant sets of finite measure for a full measure set of directions in the language of essential values we prove that the skew products which arise as poincare maps of the above systems are associated to non regular mathbb z valued cocycles for interval exchange transformations

on the growth of l 2 invariants for sequences of lattices in lie groups we study the asymptotic behaviour of betti numbers twisted torsion and other spectral invariants of sequences of locally symmetric spaces our main results are uniform versions of the degeorge wallach theorem of a theorem of delorme and various other limit multiplicity theorems a basic idea is to adapt the notion of benjamini schramm convergence bs convergence originally introduced for sequences of finite graphs of bounded degree to sequences of riemannian manifolds and analyze the possible limits we show that bs convergence of locally symmetric spaces g k implies convergence in an appropriate sense of the normalized relative plancherel measures associated to l2 g this then yields convergence of normalized multiplicities of unitary representations betti numbers and other spectral invariants on the other hand when the corresponding lie group g is simple and of real rank at least two we prove that there is only one possible bs limit i e when the volume tends to infinity locally symmetric spaces always bs converge to their universal cover g k this leads to various general uniform results when restricting to arbitrary sequences of congruence covers of a fixed arithmetic manifold we prove a strong quantitative version of bs convergence which in turn implies upper estimates on the rate of convergence of normalized betti numbers in the spirit of sarnak xue an important role in our approach is played by the notion of invariant random subgroups for higher rank simple lie groups g we exploit rigidity theory and in particular the nevo stuck zimmer theorem and kazhdan s property t to obtain a complete understanding of the space of irs s of g 2017 department of mathematics princeton university

limit sets of teichm uller geodesics with minimal nonuniquely ergodic vertical foliation ii given a sequence of curves on a surface we provide con ditions which ensure that 1 the sequence is an innite quasi geodesic in the curve complex 2 the limit in the gromov boundary is repre sented by a nonuniquely ergodic ending lamination and 3 the sequence divides into a nite set of subsequences each of which projectively con verges to one of the ergodic measures on the ending lamination the conditions are suciently robust allowing us to construct sequences on a closed surface of genus g for which the space of measures has the maximal dimension 3g 3 for example we also study the limit sets in the thurston boundary of teichmuller geodesic rays dened by quadratic dierentials whose vertical foliations are obtained from the constructions mentioned above we prove that such examples exist for which the limit is a cycle the 1 skeleton of the simplex of projective classes of measures visiting every vertex

relation between volatility correlations in financial markets and omori processes occurring on all scales we analyze the memory in volatility by studying volatility return intervals defined as the time between two consecutive fluctuations larger than a given threshold in time periods following stock market crashes such an aftercrash period is characterized by the omori law which describes the decay in the rate of aftershocks of a given size with time t by a power law with exponent close to 1 a shock followed by such a power law decay in the rate is here called omori process we find self similar features in the volatility specifically within the aftercrash period there are smaller shocks that themselves constitute omori processes on smaller scales similar to the omori process after the large crash we call these smaller shocks subcrashes which are followed by their own aftershocks we also show that the omori law holds not only after significant market crashes as shown by lillo and mantegna phys rev e 68 016119 2003 but also after intermediate shocks by appropriate detrending we remove the influence of the crashes and subcrashes from the data and find that this procedure significantly reduces the memory in the records moreover when studying long term correlated fractional brownian motion and autoregressive fractionally integrated moving average artificial models for volatilities we find omori type behavior after high volatilities thus our results support the hypothesis that the memory in the volatility is related to the omori processes present on different time scales

a matrix method for estimating the liapunov exponent of one dimensional systems let 0 1 0 1 be a piecewise monotonie expanding map then admits an absolutely continuous invariant measure a result of kosyakin and sandler shows that can be approximated by a sequence of absolutely continuous measures n invariant under piecewise linear markov maps itn each itn is constructed on the inverse images of the turning points of the easily computable measures n are used to estimate the liapunov exponent of the idea of using markov maps for estimating the liapunov exponent is applied to both expanding and nonexpanding maps

automatic algorithm for monotone trend removal the available numerical algorithms for trend removal require a direct subjective intervention in choosing critical parameters in this paper an algorithm is presented which needs no initial subjective assumptions monotone trends are approximated by piecewise linear curves obtained by dividing into subintervals the signal values interval not the time interval the slope of each linear segment of the estimated trend is proportional to the average one step displacement of the signal values included into the corresponding subinterval the evaluation of the trend removal is performed on statistical ensembles of artificial time series with the random component given by realizations of autoregressive of order one stochastic processes or by fractional brownian motions the accuracy of the algorithm is compared with that of two well tested methods polynomial fitting and a nonparametric method based on moving average for stationary noise the results of the algorithm are slightly better but for nonstationary noise the preliminary results indicate that the polynomial fitting has the best accuracy as a verification on a real time series the time periods with monotone variation of global average temperature over the last 1800 phantom rule 0 3em 0ex text years are established the removal of a nonmonotone trend is also briefly discussed

qualitative properties of systems of two complex homogeneous ode s a connection to polygonal billiards a correspondence between the orbits of a system of two complex homogeneous polynomial ordinary differential equations with real coefficients and those of a polygonal billiard is displayed this correspondence is general in the sense that it applies to an open set of systems of ordinary differential equations of the specified kind this allows us to transfer results well known from the theory of polygonal billiards such as ergodicity the existence of periodic orbits the absence of exponential divergence the existence of additional conservation laws and the presence of discontinuities in the dynamics to the corresponding systems of ordinary differential equations it also shows that the considerable intricacy known to exist for polygonal billiards also attends these apparently simpler systems of ordinary differential equations

the poisson boundary of text out f_n let mu be a probability measure on text out f n with finite first logarithmic moment with respect to the word metric finite entropy and whose support generates a nonelementary subgroup of text out f n we show that almost every sample path of the random walk on text out f n mu when realized in culler and vogtmann s outer space converges to the simplex of a free arational tree we then prove that the space mathcal fi of simplices of free and arational trees equipped with the hitting measure is the poisson boundary of text out f n mu using bestvina reynolds and hamenstadt s description of the gromov boundary of the complex mathcal ff n of free factors of f n this gives a new proof of the fact due to calegari and maher that the realization in mathcal ff n of almost every sample path of the random walk converges to a boundary point we get in addition that partial mathcal ff n equipped with the hitting measure is the poisson boundary of text out f n mu

quantum ergodicity of boundary values of eigenfunctions suppose that n is a bounded piecewise smooth domain we prove that the boundary values cauchy data of eigenfunctions of the laplacian on with various boundary conditions are quantum ergodic if the classical billiard map on the ball bundle b is ergodic our proof is based on the classical observation that the boundary values of an interior eigenfunction 2 is an eigenfunction of an operator fh on the boundary of with h 1 in the case of the neumann boundary condition fh is the boundary integral operator induced by the double layer potential we show that fh is a semiclassical fourier integral operator quantizing the billiard map plus a small remainder the quantum dynamics defined by fh can be exploited on the boundary much as the quantum dynamics generated by the wave group were exploited in the interior of domains with corners and ergodic billiards in the work of zelditch zworski 1996 novelties include the facts that fh is not unitary and consequently the boundary values are equidistributed by measures which are not invariant under and which depend on the boundary conditions ergodicity of boundary values of eigenfunctions on domains with ergodic billiards was conjectured by s ozawa 1993 and was almost simultaneously proved by gerard leichtnam 1993 in the case of convex c1 1 domains with continuous tangent planes and with dirichlet boundary conditions our methods seem to be quite different motivation to study piecewise smooth domains comes from the fact that almost all known ergodic domains are of this form

global models for the orientation field of fingerprints an approach based on quadratic differentials quadratic differentials naturally define analytic orientation fields on planar surfaces we propose to model orientation fields of fingerprints by specifying quadratic differentials models for all fingerprint classes such as arches loops and whorls are laid out these models are parametrised by few geometrically interpretable parameters which are invariant under euclidean motions we demonstrate their ability in adapting to given observed orientation fields and we compare them to existing models using the fingerprint images of the nist special database 4 we also illustrate that these model allow for extrapolation into unobserved regions this goes beyond the scope of earlier models for the orientation field as those are restricted to the observed planar fingerprint region within the framework of quadratic differentials we are able to verify analytically penrose s formula for the singularities on a palm l s penrose dermatoglyphics scientific american vol 221 no 6 pp 73 84 1969 potential applications of these models are the use of their parameters as indices of large fingerprint databases as well as the definition of intrinsic coordinates for single fingerprint images

ergodicity of eigenfunctions for ergodic billiards we give a simple proof of ergodicity of eigenfunctions of the laplacian with dirichlet boundary conditions on compact riemannian manifolds with piecewise smooth boundaries and ergodic billiards examples include the bunimovich stadium the sinai billiard and the generic polygonal billiard tables of kerckhoff masur and smillie

schrodinger equation for convex plane polygons a tiling method for the derivation of eigenvalues and eigenfunctions motivated by a recently advanced conjecture on the ergodic properties of quantum systems the problem of solving the schrodinger equation for a free particle in a plane polygonal enclosure is revisited it will be shown that two elementary lemmas suffice to give a complete characterization of the polygons for which a solution can be found in terms of a finite superposition of plane waves without making use of advanced group theoretical techniques it turns out inter alia that these polygons considered as classical billiards are all and only those which are completely integrable in the sense of arnold s theorem motivated by a recently advanced conjecture on the ergodic properties of quantum systems the problem of solving the schrodinger equation for a free particle in a plane polygonal enclosure is revisited it will be shown that two elementary lemmas suffice to give a complete characterization of the polygons for which a solution can be found in terms of a finite superposition of plane waves without making use of advanced group theoretical techniques it turns out inter alia that these polygons considered as classical billiards are all and only those which are completely integrable in the sense of arnold s theorem

interaction between deterministic trend and autoregressive process 1 when both an autoregressive ar process stochastic trend and a deterministic trend systematic changes over mean exist within a time series there may be some interactions between them this study investigates whether these two components interact with each other only time series consisting of a linear trend and a lag 1 autoregressive ar 1 process are explored which are commonly used in hydrology results indicate that 1 the presence of a deterministic trend will overestimate positive serial correlation and underestimate negative serial correlation while 2 the existence of an ar 1 process does not affect the estimate of the magnitude of the deterministic trend however a positive ar 1 will inflate the variance of the trend and a negative ar 1 will shrink the variance of the trend for a time series with a deterministic trend the trend has to be removed in order to correctly model the stochastic properties of the time series in the literature both differencing and detrending have been proposed to fulfill such a task their applicability is based on the assumption that they cannot distort the residual series however no evidence has been provided to certify whether this assumption is appropriate this study examines this issue and it indicates that differencing can remove a deterministic trend from a time series but it can seriously damage the existing ar 1 process in contrast to differencing detrending can remove a deterministic trend without distorting the existing ar process

on the cohomological dimension of the moduli space of riemann surfaces the moduli space of riemann surfaces of genus g 2 is up to a finite etale cover a complex manifold and so it makes sense to speak of its dolbeault cohomological dimension the conjecturally optimal bound is g 2 this expectation is verified in low genus and supported by harer s computation of its de rham cohomological dimension and by vanishing results in the tautological intersection ring in this paper we prove that such dimension is at most 2g 2 we also prove an analogous bound for the moduli space of riemann surfaces with marked points the key step is to show that the dolbeault cohomological dimension of each stratum of translation surfaces is at most g in order to do that we produce an exhaustion function whose complex hessian has controlled index in the construction of such a function some basic geometric properties of flat surfaces come into play

effect of nonlinear filters on detrended fluctuation analysis when investigating the dynamical properties of complex multiple component physical and physiological systems it is often the case that the measurable system s output does not directly represent the quantity we want to probe in order to understand the underlying mechanisms instead the output signal is often a linear or nonlinear function of the quantity of interest here we investigate how various linear and nonlinear transformations affect the correlation and scaling properties of a signal using the detrended fluctuation analysis dfa which has been shown to accurately quantify power law correlations in nonstationary signals specifically we study the effect of three types of transforms i linear y i a x i b ii nonlinear polynomial y i a x k i and iii nonlinear logarithmic y i log x i delta filters we compare the correlation and scaling properties of signals before and after the transform we find that linear filters do not change the correlation properties while the effect of nonlinear polynomial and logarithmic filters strongly depends on a the strength of correlations in the original signal b the power k of the polynomial filter and c the offset delta in the logarithmic filter we further apply the dfa method to investigate the apparent scaling of three analytic functions i exponential exp x a ii logarithmic log x a and iii power law x a lambda which are often encountered as trends in physical and biological processes while these three functions have different characteristics we find that there is a broad range of values for parameter a common for all three functions where the slope of the dfa curves is identical we further note that the dfa results obtained for a class of other analytic functions can be reduced to these three typical cases we systematically test the performance of the dfa method when estimating long range power law correlations in the output signals for different parameter values in the three types of filters and the three analytic functions we consider

cannon thurston maps for trees of hyperbolic metric spaces let x d be a tree t of hyperbolic metric spaces satisfying the quasi isometrically embedded condition let v be a vertex of t let x v d v denote the hyperbolic metric space corresponding to v then i x v x extends continuously to a map i x v x this generalizes a theorem of cannon and thurston the techniques are used to give a new proof of a result of minsky thurston s ending lamination conjecture for certain kleinian groups applications to graphs of hyperbolic groups and local connectivity of limit sets of kleinian groups are also given

limit sets of teichmuller geodesics with minimal non uniquely ergodic vertical foliation we describe a method for constructing teichmuller geodesics where the vertical foliation is minimal but is not uniquely ergodic and where we have a good understanding of the behavior of the teichmuller geodesic the construction depends on various parameters and we show that one can ad just the parameters to ensure that the set of accumulation points of such a geodesic in the thurston boundary is exactly the projective 1 simplex of all projective measured foliations that are topologically equivalent to with fur ther adjustment of the parameters one can further assume that the transverse measure is an ergodic measure on the non uniquely ergodic foliation

what are lyapunov exponents and why are they interesting taking as inspiration the fields medal work of artur avila i d like to introduce you to lyapunov exponents my plan is to show how lyapunov exponents play a key role in three areas in which avila s results lie smooth ergodic theory billiards and translation surfaces and the spectral theory of 1 dimensional schrodinger operators but first what are lyapunov exponents let s begin by viewing them in one of their natural habitats the barycentric subdivision of a triangle is a collection of 6 smaller triangles obtained by joining the midpoints of the sides to opposite vertices here s what happens when you start with an equilateral triangle and repeatedly barycentrically subdivide

matrices and eigenfunctions induced by markov maps abstract the matrix m induced by a markov map has 1 as the eigenvalue of maximum modulus if m is also irreducible then the algebraic and geometric multiplicities of the eigenvalue 1 are also 1 let g g 1 g 2 g n be an n row vector such that g 1 c and g i c i 2 then every such g is the unique upto constant multiples left eigenvector associated with the eigenvalue 1 of a matrix m induced by a markov map

ergodic directions for billiards in a strip with periodically located obstacles we study the size of the set of ergodic directions for the directional billiard flows on the infinite band mathbb r times 0 h with periodically placed linear barriers of length 0 h we prove that the set of ergodic directions is always uncountable moreover if h 0 1 is rational the hausdorff dimension of the set of ergodic directions is greater than 1 2 in both cases rational and irrational we construct explicitly some sets of ergodic directions

on ergodicity of foliations on mathbb z d covers of half translation surfaces and some applications to periodic systems of eaton lenses we consider the geodesic flow defined by periodic eaton lens patterns in the plane and discover ergodic ones among those the ergodicity result on eaton lenses is derived from a result for quadratic differentials on the plane that are pull backs of quadratic differentials on tori ergodicity itself is concluded for mathbb z d covers of quadratic differentials on compact surfaces with vanishing lyapunov exponents

billiards in polygons survey of recent results in the ten or so years since the publication of ref 19 polygonal billiards have remained an active subject of research in the mathematics and physics literature as a result our understanding of the subject although still far from being complete is much better than it was ten years ago this survey attempts to give a broad overview of the dynamics of billiards in polygons with an emphasis on the material that was not in the literature in 1985 this is not a survey of the publications on polygonal billiards and we apologize to authors whose work has not been included the selection of topics has been strongly influenced by the personal taste of the author and by space limitations thus we do not discuss quantum polygonal billiards in particular quantum chaos see e g refs 41 44 and 45 we hope that somebody will write a survey on this important subject in the near future a fascinating aspect of the subject is the interplay between the geometric shape of the billiard table i e a planar curve and the qualitative features of the billiard dynamics hence it is instructive to compare polygonal billiard tables with other classes of billiard tables in particular the smooth strictly convex tables and the dispersing or sinai tables produce strikingly different types of billiard dynamics we recommend refs 46 and 15 for a general overview of the subject and a comparative study of billiard dynamics for various types of billiard tables

scaling properties and entropy of long range correlated time series we calculate the shannon entropy of a time series by using the probability density functions of the characteristic sizes of the long range correlated clusters introduced in a carbone g castelli h e stanley phys rev e 69 2004 026105 we define three different measures of the entropy related respectively to the length the duration and the area of the clusters for all the three cases the entropy increases as the logarithm of a power of the size with exponents equal to the fractal dimension of the cluster length duration and area respectively r 2007 elsevier b v all rights reserved

constructive approximations to densities invariant under nonexpanding transformations let 0 1 0 1 be a piecewise monotonic nonexpanding map which has an invariant densityg and is topologically conjugate to a piecewise monotonic expanding map where the conjugacy is absolutely continuous an effective computable method is presented for approximatingg

from rational billiards to dynamics on moduli spaces this short expository note gives an elementary introduction to the study of dynamics on certain moduli spaces and in particular the recent breakthrough result of eskin mirzakhani and mohammadi we also discuss the context and applications of this result and connections to other areas of mathematics such as algebraic geometry teichmuller theory and ergodic theory on

correlations and spectra of triangular billiards abstract we consider the motion of a point particle in right triangular billiards for which acute angles are not rationally connected to we consider in particular the behaviour of correlations for a particular phase space variable the behaviour is compatible with the mixing character of the dynamics even though large fluctuations are exhibited by fourier transforming correlation functions we may as well analyse scaling properties of the spectral measure both dynamical and spectral features are quite similar to what observed in a class of quantum kicked systems

measurement of the lyapunov spectrum from a chaotic time series the exponential divergence or convergence of nearby trajectories lyapunov exponents is conceptually the most basic indicator of deterministic chaos we propose a new method to determine the spectrum of several lyapunov exponents including positive zero and even negative ones from the observed time series of a single variable we have applied the method to various known model systems and also to the rayleigh b enard experiment and have elucidated the dependence of the lyapunov exponents on the rayleigh number

weak mixing directions in non arithmetic veech surfaces we show that the billiard in a regular polygon is weak mixing in almost every invariant surface except in the trivial cases which give rise to lattices in the plane triangle square and hexagon more generally we study the problem of prevalence of weak mixing for the directional flow in an arbitrary non arithmetic veech surface and show that the hausdorff dimension of the set of non weak mixing directions is not full we also provide a necessary condition verified for instance by the veech surface corresponding to the billiard in the pentagon for the set of non weak mixing directions to have positive hausdorff dimension

hausdorff dimension of singular vectors we prove that the set of singular vectors in rd d 2 has hausdorff dimension d2d 1 and that the hausdorff dimension of the set of e dirichlet improvable vectors in rd is roughly d2d 1 plus a power of e between d2 and d as a corollary the set of divergent trajectories of the flow by diag et et e dt acting on sld 1 r sld 1 z has hausdorff codimension dd 1 these results extend the work of the first author

connectivity of the space of ending laminations we prove that for any closed surface of genus at least four and any punctured surface of genus at least two the space of ending laminations is connected a theorem of e klarreich 28 theorem 1 3 implies that this space is homeomorphic to the gromov boundary of the complex of curves it follows that the boundary of the complex of curves is connected in these cases answering the conjecture of p storm other applications include the rigidity of the complex of curves and connectivity of spaces of degenerate kleinian groups

optimal shape and location of sensors for parabolic equations with random initial data in this article we consider parabolic equations on a bounded open connected subset omega of mathbb r n we model and investigate the problem of optimal shape and location of the observation domain having a prescribed measure this problem is motivated by the question of knowing how to shape and place sensors in some domain in order to maximize the quality of the observation for instance what is the optimal location and shape of a thermometer we show that it is relevant to consider a spectral optimal design problem corresponding to an average of the classical observability inequality over random initial data where the unknown ranges over the set of all possible measurable subsets of omega of fixed measure we prove that under appropriate sufficient spectral assumptions this optimal design problem has a unique solution depending only on a finite number of modes and that the optimal domain is semi analytic and thus has a finite number of connected components this result is in strong contrast with hyperbolic conservative equations wave and schrodinger studied in privat et al j eur math soc 2015 for which relaxation does occur we also provide examples of applications to anomalous diffusion or to the stokes equations in the case where the underlying operator is any positive possible fractional power of the negative of the dirichlet laplacian we show that surprisingly enough the complexity of the optimal domain may strongly depend on both the geometry of the domain and on the positive power the results are illustrated with several numerical simulations

periodic billiard orbits are dense in rational polygons we show that periodic orbits are dense in the phase space for billiards in polygons for which the angle between each pair of sides is a rational multiple of 7r

teichmuller curves triangle groups and lyapunov exponents we construct a teichmuller curve uniformized by a fuchsian triangle group commensurable to m n for every m n in most cases for example when m 6 n and m or n is odd the uniformizing group is equal to the triangle group m n our construction includes the teichmuller curves constructed by veech and ward as special cases the construction essentially relies on properties of hypergeometric differential operators for small m we find billiard tables that generate these teich muller curves we interpret some of the so called lyapunov exponents of the kontsevich zorich cocycle as normalized degrees of a natural line bundle on a teichmuller curve we determine the lyapunov exponents for the teichmuller curves we construct

statistical analysis and spectral estimation techniques for one dimensional chaotic signals signals arising out of nonlinear dynamics are compelling models for a wide range of both natural and man made phenomena in contrast to signals arising out of linear dynamics extremely rich behavior is obtained even when we restrict our attention to one dimensional 1 d chaotic systems with certain smoothness constraints an important class of such systems are the so called markov maps we develop several properties of signals obtained from markov maps and present analytical techniques for computing a broad class of their statistics in closed form these statistics include for example correlations of arbitrary order and all moments of such signals among several results we demonstrate that all markov maps produce signals with rational spectra and we can therefore view the associated signals as chaotic arma processes with chaotic white noise as a special case finally we also demonstrate how markov maps can be used to approximate to arbitrary accuracy the statistics any of a broad class of non markov chaotic maps

geometry of the mapping class groups i boundary amenability we construct a geometric model for the mapping class group mathcal m mathcal c mathcal g of a non exceptional oriented surface s of genus g with k punctures and use it to show that the action of mathcal m mathcal c mathcal g on the compact metrizable hausdorff space of complete geodesic laminations for s is topologically amenable as a consequence the novikov higher signature conjecture holds for every subgroup of mathcal m mathcal c mathcal g

critical scale invariance in a healthy human heart rate we demonstrate the robust scale invariance in the probability density function pdf of detrended healthy human heart rate increments which is preserved not only in a quiescent condition but also in a dynamic state where the mean level of the heart rate is dramatically changing this scale independent and fractal structure is markedly different from the scale dependent pdf evolution observed in a turbulentlike cascade heart rate model these results strongly support the view that a healthy human heart rate is controlled to converge continually to a critical state

effect of trends on detrended fluctuation analysis scaling behavior we find that crossovers result from the competition between the scaling of the noise and the apparent scaling of the trend we study how the characteristics of these crossovers depend on i the slope of the linear trend ii the amplitude and period of the periodic trend iii the amplitude and power of the power law trend and iv the length as well as the correlation properties of the noise surprisingly we find that the crossovers in the scaling of noisy signals with trends also follow scaling laws i e long range power law dependence of the position of the crossover on the parameters of the trends we show that the dfa result of noise with a trend can be exactly determined by the superposition of the separate results of the dfa on the noise and on the trend assuming that the noise and the trend are not correlated if this superposition rule is not followed this is an indication that the noise and the superposed trend are not independent so that removing the trend could lead to changes in the correlation properties of the noise in addition we show how to use dfa appropriately to minimize the effects of trends how to recognize if a crossover indicates indeed a transition from one type to a different type of underlying correlation or if the crossover is due to a trend without any transition in the dynamical properties of the noise

the farrell jones conjecture for mapping class groups we prove the farrell jones conjecture for mapping class groups the proof uses the masur minsky theory of the large scale geometry of mapping class groups and the geometry of the thick part of teichmuller space the proof is presented in an axiomatic setup extending the projection axioms in bestvina et al publ math inst hautes etudes sci 122 1 64 2015 more specifically we prove that the action of mathrm mod sigma on the thurston compactification of teichmuller space is finitely mathcal f amenable for the family mathcal f consisting of virtual point stabilizers

characterizations of lattice surfaces we answer a question of vorobets by showing that the lattice property for flat surfaces is equivalent to the existence of a positive lower bound for the areas of affine triangles we show that the set of affine equivalence classes of lattice surfaces with a fixed positive lower bound for the areas of triangles is finite and we obtain explicit bounds on its cardinality we deduce several other characterizations of the lattice property

the growth rate for the number of singular and periodic orbits for a polygonal billiard for any simply connected polygon in the plane the number of billiard orbits which begin and end at a vertex grows subexponentially with respect to the length or to the number of reflections this implies that the numbers of isolated periodic orbits and of families of parallel periodic orbits do grow subexponentially the main technical device is a calculation showing that the topological entropy of the poincare map for the billiard flow is equal to zero

detecting long range correlations with detrended fluctuation analysis abstract we examine the detrended fluctuation analysis dfa which is a well established method for the detection of long range correlations in time series we show that deviations from scaling which appear at small time scales become stronger in higher orders of dfa and suggest a modified dfa method to remove them the improvement is necessary especially for short records that are affected by non stationarities furthermore we describe how crossovers in the correlation behavior can be detected reliably and determined quantitatively and show how several types of trends in the data affect the different orders of dfa

ergodicity of billiard flows and quadratic differentials on considere un systeme billard de 2 objets de masses m 1 et m 2 on montre que pour un ensemble dense de paires m 1 m 2 ce systeme est ergodique

singular perturbations of piecewise monotonic maps of the interval let t 0 1 0 1 be a piecewise monotonic c2 and expanding map in computing an orbit i x0 i 0 we model the roundoff error at each iteration by a singular perturbation i e xn 1 xn w wherew is a random variable taking on discrete values in an interval e e the main result proves that this process admits an absolutely continuous invariant measure which approaches the absolutely continuous measure invariant under the deterministic map t as the precision of computation e 0

weak mixing for interval exchange transformations and translation flows we prove that a typical interval exchange transformation is either weakly mixing or it is an irrational rotation we also conclude that a typical translation flow on a typical translation surface of genus g 2 with prescribed singularity types is weakly mixing

hausdorff dimension of the set of singular pairs in this paper we show that the hausdorff dimension of the set of singular pairs is 4 3 we also show that the action of diag e t e t e 2t on sl 3 r sl 3 z admits divergent trajectories that exit to infinity at arbitrarily slow prescribed rates answering a question of a n starkov as a by product of the analysis we obtain a higher dimensional generalization of the basic inequalities satisfied by convergents of continued fractions as an illustration of the technique used to compute hausdorff dimension we reprove a result of i j good asserting that the hausdorff dimension of the set of real numbers with divergent partial quotients is 1 2

closed trajectories for quadratic differentials with an application to billiards on considere les orbites periodiques pour le systeme dynamique d une table polygonale avec des angles multiples rationnels de le billard rationnel

a theorem on ergodicity of two dimensional hyperbolic billiards ergodicity of two dimensional billiards which satisfy some general conditions is proved this theorem is applied to one concrete class of billiards that contains in particular billiards in the stadium

the cohomological equation for roth type interval exchange maps 0 introduction 0 1 interval exchange maps 0 2 the cohomological equation 0 3 summary of the contents 1 the continued fraction algorithm for interval exchange maps 1 1 interval exchange maps 1 2 the continued fraction algorithm 1 3 roth type interval exchange maps 2 the cohomological equation 2 1 the theorem of gottschalk and hedlund 2 2 special birkhoff sums 2 3 estimates for functions of bounded variation 2 4 primitives of functions of bounded variation 3 suspensions of interval exchange maps 3 1 suspension data 3 2 construction of a riemann surface 3 3 compactification of m 3 4 the cohomological equation for higher smoothness 4 proof of full measure for roth type 4 1 the basic operation of the algorithm for suspensions 4 2 the teichmuller flow 4 3 the absolutely continuous invariant measure 4 4 integrability of log z 1 4 5 conditions b and c have full measure 4 6 the main step 4 7 condition a has full measure 4 8 proof of the proposition appendix a roth type conditions in a concrete family of interval exchange maps appendix b a nonuniquely ergodic interval exchange map satisfying condition a

analysis of clusters formed by the moving average of a long range correlated time series we analyze the stochastic function c n i ensuremath equiv y i ensuremath y n i where y i is a long range correlated time series of length n mathrm max and y n i ensuremath equiv 1 n ensuremath sum k 0 n ensuremath 1 y i ensuremath k is the moving average with window n we argue that c n i generates a stationary sequence of self affine clusters mathcal c with length mathcal l lifetime ensuremath tau and area s the length and the area are related to the lifetime by the relationships mathcal l ensuremath sim ensuremath tau ensuremath psi mathcal l and s ensuremath sim ensuremath tau ensuremath psi s where ensuremath psi mathcal l 1 and ensuremath psi s 1 h we also find that mathcal l ensuremath tau and s are power law distributed with exponents depending on h p mathcal l ensuremath sim mathcal l ensuremath ensuremath alpha p ensuremath tau ensuremath sim ensuremath tau ensuremath ensuremath beta and p s ensuremath sim s ensuremath ensuremath gamma with ensuremath alpha ensuremath beta 2 ensuremath h and ensuremath gamma 2 1 h these predictions are tested by extensive simulations on series generated by the midpoint displacement algorithm of assigned hurst exponent h ranging from 0 05 to 0 95 of length up to n mathrm max 2 21 and n up to 2 13

on the bernoulli property of planar hyperbolic billiards we consider billiards in non polygonal domains of the plane with boundary consisting of curves of three different types straight segments strictly convex inward curves and strictly convex outward curves of a special kind the billiard map for these domains is known to have non vanishing lyapunov exponents a e provided that the distance between the curved components of the boundary is sufficiently large and the set of orbits having collisions only with the flat part of the boundary has zero measure under a few additional conditions we prove that there exists a full measure set of the billiard phase space such that each of its points has a neighborhood contained up to a zero measure set in one bernoulli component of the billiard map using this result we show that there exists a large class of planar hyperbolic billiards that have the bernoulli property this class includes the billiards in convex domains bounded by straight segments and strictly convex inward arcs constructed by donnay

deviation of ergodic averages for rational polygonal billiards we prove a polynomial upper bound on the deviation of er godic averages for almost all directional flows on every tran slation sur face in particular for the generic directional flow of bill iards in any euclidean polygon with rational angles

criticality and phase transition in stock price fluctuations we analyze the behavior of the u s s p 500 index from 1984 to 1995 and characterize the non gaussian probability density functions pdf of the log returns the temporal dependence of fat tails in the pdf of a ten minute log return shows a gradual systematic increase in the probability of the appearance of large increments on approaching black monday in october 1987 reminiscent of parameter tuning towards criticality on the occurrence of the black monday crash this culminates in an abrupt transition of the scale dependence of the non gaussian pdf towards scale invariance characteristic of critical behavior these facts suggest the need for revisiting the turbulent cascade paradigm recently proposed for modeling the underlying dynamics of the financial index to account for time varying phase transitionlike and scale invariant critical like behavior

serial correlation of detrended time series a preliminary essential procedure in time series analysis is the separation of the deterministic component from the random one if the signal is the result of superposing a noise over a deterministic trend then the first one must estimate and remove the trend from the signal to obtain an estimation of the stationary random component the errors accompanying the estimated trend are conveyed as well to the estimated noise taking the form of detrending errors therefore the statistical errors of the estimators of the noise parameters obtained after detrending are larger than the statistical errors characteristic to the noise considered separately in this paper we study the detrending errors by means of a monte carlo method based on automatic numerical algorithms for nonmonotonic trends generation and for construction of estimated polynomial trends alike to those obtained by subjective methods for a first order autoregressive noise we show that in average the detrending errors of the noise parameters evaluated by means of the autocovariance and autocorrelation function are almost uncorrelated to the statistical errors intrinsic to the noise and they have comparable magnitude for a real time series with significant trend we discuss a recursive method for computing the errors of the estimated parameters after detrending and we show that the detrending error is larger than the half of the total error

ergodic theorem ergodic theory and statistical mechanics this perspective highlights the mean ergodic theorem established by john von neumann and the pointwise ergodic theorem established by george birkhoff proofs of which were published nearly simultaneously in pnas in 1931 and 1932 these theorems were of great significance both in mathematics and in statistical mechanics in statistical mechanics they provided a key insight into a 60 y old fundamental problem of the subject namely the rationale for the hypothesis that time averages can be set equal to phase averages the evolution of this problem is traced from the origins of statistical mechanics and boltzman s ergodic hypothesis to the ehrenfests quasi ergodic hypothesis and then to the ergodic theorems we discuss communications between von neumann and birkhoff in the fall of 1931 leading up to the publication of these papers and related issues of priority these ergodic theorems initiated a new field of mathematical research called ergodic theory that has thrived ever since and we discuss some of recent developments in ergodic theory that are relevant for statistical mechanics

assessment of long range correlation in time series how to avoid pitfalls due to the ubiquity of time series with long range correlation in many areas of science and engineering analysis and modeling of such data is an important problem while the field seems to be mature three major issues have not been satisfactorily resolved i many methods have been proposed to assess long range correlation in time series under what circumstances do they yield consistent results ii the mathematical theory of long range correlation concerns the behavior of the correlation of the time series for very large times a measured time series is finite however how can we relate the fractal scaling break at a specific time scale to important parameters of the data iii an important technique in assessing long range correlation in a time series is to construct a random walk process from the data under the assumption that the data are like a stationary noise process due to the difficulty in determining whether a time series is stationary or not however one cannot be 100 sure whether the data should be treated as a noise or a random walk process is there any penalty if the data are interpreted as a noise process while in fact they are a random walk process and vice versa in this paper we seek to gain important insights into these issues by examining three model systems the autoregressive process of order 1 on off intermittency and l evy motions and considering an important engineering problem target detection within sea clutter radar returns we also provide a few rules of thumb to safeguard against misinterpretations of long range correlation in a time series and discuss relevance of this study to pattern recognition

teichmuller curves in moduli space eisenstein series and an application to triangular billiards there exists a teichmuller disc n containing the riemann surface ofy2 x n 1 in the genus n 1 2 teichmuller space such that the stabilizer of n in the mapping class group has a fundamental domain of finite poincare volume in n application is given to an asymptotic formula for the length spectrum of the billiard in isosceles triangles with angles n n n 2 n and to the uniform distribution of infinite billiard trajectories in the same triangles

computing the topological entropy of general one dimensional maps a matrix theoretic method for computing the topological entropy of continuous piecewise monotonic maps of the interval is presented the method results in a constructive procedure which is easily implemented on the computer examples for families of unimodal nonunimodal and discontinuous maps are presented

dichotomy for the hausdorff dimension of the set of nonergodic directions given an irrational 0 1 we consider billiards in the table p formed by a tfrac 1 2 times1 rectangle with a horizontal barrier of length frac 1 lambda 2 with one end touching at the midpoint of a vertical side let ne p be the set of such that the flow on p in direction is not ergodic we show that the hausdorff dimension of ne p can only take on the values 0 and tfrac 1 2 depending on the summability of the series sum k frac log log q k 1 q k where q k is the sequence of denominators of the continued fraction expansion of more specifically we prove that the hausdorff dimension is frac 1 2 if this series converges and 0 otherwise this extends earlier results of boshernitzan and cheung

billiards and teichmuller curves on hilbert modular surfaces this paper exhibits an infinite collection of algebraic curves isometrically embedded in the moduli space of riemann surfaces of genus two these teichmuller curves lie on hilbert modular surfaces parameterizing abelian varieties with real multiplication explicit examples constructed from l shaped polygons give billiard tables with optimal dynamical properties