limiting distribution of translates of the orbit of a maximal pmb mathbb q q torus from identity on mathrm sl _n pmb mathbb r mathrm sl _n pmb mathbb z sl n r sl n z consider the orbit of real points of a maximal mathbb q torus from identity coset on mathrm sl n mathbb r mathrm sl n mathbb z it naturally carries a possibly infinite haar measure we classify all possible limit measures of it when translated by a sequence of elements from mathrm sl n mathbb r this is a natural extension of shapira and zheng s work where only mathbb q split tori are considered

thermodynamics dimension and the weil petersson metric in this paper we study branched coverings of metrized simplicial trees f t t which arise from polynomial maps f c c with disconnected julia sets we show that the collection of all such trees up to scale forms a contractible space ptd compactifying the moduli space of polynomials of degree d that f records the asymptotic behavior of the multipliers of f and that any meromorphic family of polynomials over can be completed by a unique tree at its central fiber in the cubic case we give a combinatorial enumeration of the trees that arise and show that pt3 is itself a tree resume dans ce travail nous etudions des revetements ramifies d arbres metriques simpliciaux f t t qui sont obtenus a partir d applications polynomiales f c c possedant un ensemble de julia non connexe nous montrons que la collection de tous ces arbres a un facteur d echelle pres forme un espace contractile ptd qui compactifier l espace des modules des polynomes de degre d nous montrons aussi que f enrigistre le comportement asymptotique des multiplicateurs de f et que tout famille meromorphe de polynomes definis sur peut etre completee par un unique arbre comme sa fibre centrale dans le cas cubique nous donnons une enumeration combinatoire des arbres ainsi obtenus et montrons que pt3 est lui meme un arbre

pair correlation densities of inhomogeneous quadratic forms under explicit diophantine conditions on 2 w eprove that the local two point correlations of the sequence given by the values m 2 n 2 with m n 2 are those of a poisson process this partly confirms a conjecture of berry and tabor 2 on spectral statistics of quantized integrable systems and also establishes a particular case of the quantitative version of the oppenheim conjecture for inhomogeneous quadratic forms of signature 2 2 the proof uses theta sums and ratner s classification of measures invariant under unipotent flows

on the growth of nonuniform lattices in pinched negatively curved manifolds we study the relation between the exponential growth rate of volume in a pinched negatively curved manifold and the critical exponent of its lattices these objects have a long and interesting story and are closely related to the geometry and the dynamical properties of the geodesic flow of the manifold

equidistribution of expanding translates of curves and diophantine approximation on matrices we study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space g gamma of a semisimple algebraic group g we define two families of algebraic subvarieties of the associated partial flag variety g p which give the obstructions to non divergence and equidistribution we apply this to prove that for lebesgue almost every point on an analytic curve in the space of m times n real matrices whose image is not contained in any subvariety coming from these two families dirichlet s theorem on simultaneous diophantine approximation cannot be improved the proof combines geometric invariant theory ratner s theorem on measure rigidity for unipotent flows and linearization technique

immersing almost geodesic surfaces in a closed hyperbolic three manifold let m 3 be a closed hyperbolic three manifold we construct closed surfaces that map by immersions into m 3 so that for each one the corresponding mapping on the universal covering spaces is an embedding or in other words the corresponding induced mapping on fundamental groups is an injection

representations of integers by an invariant polynomial and unipotent flows we study a refined version of linnik s problem on the asymptotic behavior of the number of representations of integers m by an integral polynomial as m tends to infinity assuming that the polynomials arise from invariant theory we reduce the question to the study of limiting behavior of measures invariant under unipotent flows our main tool is then ratner s theorem on the uniform distribution of unipotent flows in a form refined by dani and margulis dm2

equidistribution of expanding translates of curves and dirichlet s theorem on diophantine approximation we show that for almost all points on any analytic curve on k which is not contained in a proper affine subspace the dirichlet s theorem on simultaneous approximation as well as its dual result for simultaneous approximation of linear forms cannot be improved the result is obtained by proving asymptotic equidistribution of evolution of a curve on a strongly unstable leaf under certain partially hyperbolic flow on the space of unimodular lattices in k 1 the proof involves ratner s theorem on ergodic properties of unipotent flows on homogeneous spaces

homogeneous asymptotic limits of haar measures of semisimple linear groups and their lattices we show that haar measures of connected semisimple groups embedded via a representation into a matrix space have a homogeneous asymptotic limit when viewed from far away and appropriately rescaled this is still true if the haar measure of the semisimple group is replaced by the haar measure of a irreducible lattice of the group and the asymptotic measure is the same in the case of an almost simple group of rank greater than 2 a remainder term is also obtained this extends and makes precise anterior results of duke rudnick and sarnak and eskin mcmullen in the case of a group variety

on the representation of integers by the lorentzian quadratic form in this paper we establish an asymptotic formula for large radiusr for the number of representations of a nonzero integerkby the lorentzian quadratic formx21 x22 x2n x2n 1that are contained in the ball of radiusrcentered at the origin in euclidean n 1 space

counting integral points in certain homogeneous spaces the leading term of asymptotic formula of the number of integral points in non compact symmetric homogeneous spaces of semi simple simply connected algebraic groups is given by the average of the product of the number of local solutions twisted by the brauer manin obstruction the similar result is also true for homogeneous spaces of reductive groups with some restriction as application we will give the explicit asymptotic formulae of the number of integral points of certain norm equations and prove the leading term of asymptotic formula of the number of integral matrices with a fixed irreducible characteristic polynomial over z studied by eskin mozes shah is equal to the product of the number of local integral solutions over all primes although the density function defined by borovoi and rudnick is not trivial in general we also answer a question raised by borovoi and rudnick for comparing the number of integral symmetric matrices with the given determinant with the product of local densities c 2015 elsevier all rights reserved

poincare series for non riemannian locally symmetric spaces abstract we initiate the spectral analysis of pseudo riemannian locally symmetric spaces g h beyond the classical cases where h is compact automorphic forms or is trivial analysis on symmetric spaces for any non riemannian reductive symmetric space x g h on which the discrete spectrum of the laplacian is nonempty and for any discrete group of isometries whose action on x is sufficiently proper we construct l 2 eigenfunctions of the laplacian on x x for an infinite set of eigenvalues these eigenfunctions are obtained as generalized poincare series i e as projections to x of sums over the orbits of eigenfunctions of the laplacian on x we prove that the poincare series we construct still converge and define nonzero l 2 functions after any small deformation of inside g for a large class of groups thus the infinite set of eigenvalues we construct is stable under small deformations this contrasts with the classical setting where the nonzero discrete spectrum varies on the teichmuller space of a compact riemann surface we actually construct joint l 2 eigenfunctions for the whole commutative algebra of invariant differential operators on x

walks on groups counting reducible matrices polynomials and surface and free group automorphisms we prove sharp limit theorems on random walks on graphs with values in finite groups we then apply these re sults together with some elementary algebraic geometry number theory and representation theory to finite quotients of lattices in semisimple lie groups specifically sl n z and sp 2n z to show that a random element in one of these lattices has irre ducible characteristic polynomials overz the term random can be defined in at least two ways in terms of height and also in terms of word length in terms of a generating set we show the result using both definitions we use the above results to show that a random in terms of word length element of the mapping class group of a surface is pseudo anosov and that a a random free group automorphism is irreducible with irreducible powers or strongly irreducible 1

invariant measures for unipotent translations on homogeneous spaces abstract it is shown that all borel probability measures invariant under unipotent translations on homogeneous spaces of connected lie groups are algebraic

semiclassical behaviour of expectation values in time evolved lagrangian states for large times we study the behaviour of time evolved quantum mechanical expectation values in lagrangian states in the limit 0 and t we show that it depends strongly on the dynamical properties of the corresponding classical system if the classical system is strongly chaotic i e anosov then the expectation values tend to a universal limit this can be viewed as an analogue of mixing in the classical system if the classical system is integrable then the expectation values need not converge and if they converge their limit depends on the initial state an additional difference occurs in the timescales for which we can prove this behaviour in the chaotic case we get up to ehrenfest time t ln 1 whereas for integrable system we have a much larger time range

some counting questions for matrices with restricted entries we obtain upper bounds for the number of arbitrary and symmetric matrices with integer entries in a given box in an arbitrary location and a given determinant we then apply these bounds to estimate the number of matrices in such boxes which have an integer eigenvalues finally we outline some open questions

effective equidistribution of shears and applications a shear is a unipotent translate of a cuspidal geodesic ray in the quotient of the hyperbolic plane by a non uniform discrete subgroup of text psl 2 mathbb r possibly of infinite co volume we prove the regularized equidistribution of shears under large translates with effective that is power saving rates we also give applications to weighted second moments of gl 2 automorphic l functions and to counting lattice points on locally affine symmetric spaces

effective equidistribution for closed orbits of semisimple groups on homogeneous spaces we prove effective equidistribution with polynomial rate for large closed orbits of semisimple groups on homogeneous spaces under certain technical restrictions notably the acting group should have finite centralizer in the ambient group the proofs make extensive use of spectral gaps and also of a closing lemma for such actions

power free values of polynomials on symmetric varieties given a symmetric variety y defined over q and a non zero polynomial with integer coefficients we use techniques from homogeneous dynamics to establish conditions under which the polynomial can be made r free for a zariski dense set of integral points on y we also establish an asymptotic counting formula for this set in the special case that y is a quadric hypersurface we give explicit bounds on the size of r by combining the argument with a uniform upper bound for the density of integral points on general affine quadrics defined over q

dynamics of sl2 ℝ over moduli space in genus two this paper classifies orbit closures and invariant measures for the natural action of sl2 r on um2 the bundle of holomorphic 1 forms over the moduli space of riemann surfaces of genus two

on raghunathan s measure conjecture this paper represents the last in our three part series on raghunathan s measure conjecture see r1 r2 for parts i and ii more specifically let g be a real lie group all groups in this paper are assumed to be second countable with the lie algebra r a discrete subgroup of g and ir g r g the projection lwg rg the group g acts by right translations on r g x x g x e r g g e g let gt be a borel probability measure on r g define

integral points of bounded height on compactifications of semi simple groups we study the asymptotic distribution of integral points of bounded height on partial bi equivariant compactifications of semi simple groups of adjoint type

asymptotic estimates for the spatial segregation of competitive systems abstract for a class of population models of competitive type we study the asymptotic behavior of the positive solutions as the competition rate tends to infinity we show that the limiting problem is a remarkable system of differential inequalities which defines the functional class s in 2 by exploiting the regularity theory recently developed in conti et al indiana univ math j to appear for the elements of functional classes of the form s we provide some qualitative and regularity property of the limiting configurations besides for the case of two competing species we obtain a full description of the limiting states and we prove some quantitative estimates for the rate of convergence finally we prove some new liouville type results which allow to have uniform regularity estimates of the solutions

lyapunov exponents for surface group representations let rho lambda lambda in lambda be a holomorphic family of representations of a surface group pi 1 s into mathrm psl 2 mathbb c where s is a topological possibly punctured surface with negative euler characteristic given a structure of riemann surface of finite type on s we construct a bifurcation current on the parameter space that is a 1 1 positive closed current attached to the bifurcations of the family it is defined as the dd c of the lyapunov exponent of the representation with respect to the brownian motion on the riemann surface s endowed with its poincare metric we show that this bifurcation current describes the asymptotic distribution of various codimension 1 phenomena in for instance the random hypersurfaces of defined by the condition that a random closed geodesic on s is mapped under to a parabolic element or the identity are asymptotically equidistributed with respect to the bifurcation current the proofs are based on our previous work deroin and dujardin invent math 190 57 118 2012 and on a careful control of a discretization procedure of the brownian motion

onsager relations and eulerian hydrodynamic limit for systems with several conservation laws we present the derivation of the hydrodynamic limit under eulerian scaling for a general class of one dimensional interacting particle systems with two or more conservation laws following yau s relative entropy method it turns out that in case of more than one conservation laws in order that the system exhibit hydrodynamic behaviour some particular identities reminiscent of onsager s reciprocity relations must hold we check validity of these identities whenever a stationary measure with product structure exists it also follows that as a general rule the equilibrium thermodynamic entropy as function of the densities of the conserved variables is a globally convex lax entropy of the hyperbolic systems of conservation laws arising as hydrodynamic limit as concrete examples we also present a number of models modeling deposition or domain growth phenomena the onsager relations arising in the context of hydrodynamic limits under hyperbolic scaling seem to be novel the fact that equilibrium thermodynamic entropy is lax entropy for the arising euler equations was noticed earlier in the context of hamiltonian systems with weak noise see ref 7

analogs of wiener s ergodic theorems for semisimple lie groups ii let g be a connected finite center simple lie group of real rank one and let k be a maximal compact subgroup of g let a at i t c r be a 1 parameter subgroup of hyperbolic translations such that g ka k is a cartan decomposition denote by ot the probability measure on g given by ot mk 6at mk where mk is the normalized haar measure on k let it denote the probablity measure with density vol bt 1xb g where bt g d gk k t and d denotes the canonical g invariant riemannian metric on g k let x b a be a standard borel probability space on which g acts measurably and ergodically by measure preserving transformations let 7r g iso lp x be the associated isometric representation and let 7r ut and 7r 3t denote the operators canonically associated with ct and it let n g dimr g k we prove maximal and pointwise ergodic theorems for the ball averages it on simple real rank one lie groups which are analogous to n wiener s maximal and pointwise ergodic theorems for ball averages on in the general case of ball averages on semisimple groups is completed in m n s our main result is the following

khinchin s theorem for approximation by integral points on quadratic varieties we prove an analogue of the khinchin s theorem for the diophantine approximation by integer vectors lying on a quadratic variety the proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group we show that in this system generic trajectories of a certain geodesic flow visit a family of shrinking subsets infinitely often

counting lattice points for a locally compact second countable group g and a lattice subgroup we give an explicit quantitative solution of the lattice point counting problem in general domains in g provided that i g has finite upper local dimension and the domains satisfy a basic regularity condition ii the mean ergodic theorem for the action of g on g holds with a rate of convergence the error term we establish matches the best current result for balls in symmetric spaces of simple higher rank lie groups but holds in much greater generality a significant advantage of the ergodic theoretic approach we use is that the solution to the lattice point counting problem is uniform over families of lattice subgroups provided they admit a uniform spectral gap in particular the uniformity property holds for families of finite index subgroups satisfying a quantitative variant of property we discuss a number of applications including counting lattice points in general domains in semisimple s algebraic groups counting rational points on group varieties with respect to a height function and quantitative angular or conical equidistribution of lattice points in symmetric spaces and in affine symmetric varieties we note that the mean ergodic theorems which we establish are based on spectral methods including the spectral transfer principle and the kunze stein phenomenon we formulate and prove appropriate analogues of both of these results in the set up of adele groups and they constitute a necessary step in our proof of quantitative results in counting rational points

translate of horospheres and counting problems let g be a semisimple lie group without compact factors gamma be an irreducible lattice in g in the first part of the article we give the necessary and sufficient condition under which a sequence of translates of probability horospherical measures is convergent and the limiting measure is also determined when it is convergent see theorems 1 and 2 for the precise statements in the second part two applications are presented the first one is of geometric nature and the second one gives an alternative way to count the number of rational points on a flag variety

liouville theorems and 1 dimensional symmetry for solutions of an elliptic system modelling phase separation abstract we consider solutions of the competitive elliptic system s u i j i u i u j 2 in r n u i 0 in r n i 1 k we are concerned with the classification of entire solutions according to their growth rate the prototype of our main results is the following there exists a function k n n increasing in k such that if u 1 u k is a solution of s and u 1 x u k x c 1 x d for every x r n then d this means that the number of components k of the solution imposes a lower bound increasing in k on the minimal growth of the solution itself if n 2 the expression of is explicit and optimal while in higher dimension it can be characterised in terms of an optimal partition problem we discuss the sharpness of our results and as a further step for every n 2 we can prove the 1 dimensional symmetry of the solutions of s satisfying suitable assumptions extending known results which are available for k 2 the proofs rest upon a blow down analysis and on some monotonicity formulae

the distribution of free path lengths in the periodic lorentz gas and related lattice point problems the periodic lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers and is one of the fundamental models for chaotic diffusion in the present paper we investigate the boltzmann grad limit where the radius of each scatterer tends to zero and prove the existence of a limiting distribution for the free path length we also discuss related problems such as the statistical distribution of directions of lattice points that are visible from a fixed position

hardy littlewood varieties and semisimple groups we are interested in counting integer and rational points in affine algebraic varieties also under congruence conditions we introduce the notions of a strongly hardy littlewood variety and a relatively hardy littlewood variety in terms of counting rational points satisfying congruence conditions the definition of a strongly hardy littlewood variety is given in such a way that varieties for which the hardy littlewood circle method is applicable are strongly hardy littlewood

equidistribution and counting for orbits of geometrically finite hyperbolic groups let g be the identity component of so n 1 n 2 acting linearly on a finite dimensional real vector space v consider a vector w0 v such that the stabilizer of w0 is a symmetric subgroup of g or the stabilizer of the line rw0 is a parabolic subgroup of g for any non elementary discrete subgroup of g with its orbit w0 discrete we compute an asymptotic formula as t for the number of points in w0 of norm at most t provided that the bowen margulis sullivan measure on t1 hn and the skinning size of w0 are finite the main ergodic ingredient in our approach is the description for the limit ing distribution of the orthogonal translates of a totally geodesically immersed closed submanifold of hn we also give a criterion on the finiteness of the skinning size of w0 for geometrically finite mathematics department brown university providence ri 02912 and korea insti tute for advanced study seoul korea e mail address heeoh math brown edu department of mathematics the ohio state university columbus oh 43210 e mail address shah math osu edu

geodesics with one self intersection and other stories abstract in this paper we show that for any hyperbolic surface s the number of geodesics of length bounded above by l in the mapping class group orbit of a fixed closed geodesic with a single double point is asymptotic to l dim teichmuller space of s since closed geodesics with one double point fall into a finite number of mod s orbits we get the same asymptotic estimate for the number of such geodesics of length bounded by l and systems of curves where one curve has a self intersection or there are two curves intersecting once we also use our elementary methods to do a more precise study of geodesics with a single double point on a punctured torus including an extension of mcshane s identity to such geodesics in the second part of the paper we study the question of when a covering of the boundary of an oriented surface s can be extended to a covering of the surface s itself we obtain a complete answer to that question and also to the question of when we can further require the extension to be a regular covering of s we also analyze the question of the minimal index of a subgroup in a surface group which does not contain a given element we show that we have a linear bound for the index of an arbitrary subgroup a cubic bound for the index of a normal subgroup but also poly log bounds for each fixed level in the lower central series using elementary arithmetic considerations the results hold for free groups and fundamental groups of closed surfaces

the good pants homology and the ehrenpreis conjecture we develop the notion of the good pants homology and show that it agrees with the standard homology on closed surfaces good pants are pairs of pants whose cus have the length nearly equal to some large number r 0 combined with our previous work on the surface subgroup theorem this yields a proof of the ehrenpreis conjecture

integral points on symmetric varieties and satake compatifications let v be an affine symmetric variety defined over q we compute the asymptotic distribution of the angular components of the integral points in v this distribution is described by a family of invariant measures concentrated on the satake boundary of v in the course of the proof we describe the structure of the satake compactifications for general affine symmetric varieties and compute the asymptotic of the volumes of norm balls

orbits of discrete subgroups on a symmetric space and the furstenberg boundary let x be a symmetric space of noncompact type and let be a lattice in the isometry group of x we study the distribution of orbits of acting on the symmetric space x and its geometric boundary x generalizing the main equidistribution result of margulis s thesis m theorem 6 to higher rank symmetric spaces more precisely for any y x and b x we investigate the distribution of the set y b 1 in x x it is proved in particular that the orbits of in the furstenberg boundary are equidistributed and that the orbits of in x are equidistributed in sectors defined with respect to a cartan decomposition our main tools are the strong wavefront lemma and the equidistribution of solvable flows on homogeneous spaces which we obtain using shah s result s corollary 1 2 based on ratner s measure classification theorem r1 theorem 1

on the uniform equidistribution of long closed horocycles it is well known that on any given hyperbolic surface of finite area a closed horocycle of length becomes asymptotically equidistributed as in this paper we prove that any subsegment of length greater than 1 2 e of such a closed horocycle also becomes equidistributed as the exponent 1 2 e is the best possible and improves upon a recent result by hejhal he3 we give two proofs of the above result our second proof leads to explicit information on the rate of convergence we also prove a result on the asymptotic joint equidistribution of a finite number of distinct subsegments having equal length proportional to

spectral form factors of rectangle billiards the berry tabor conjecture asserts that local statistical measures of the eigenvalues j of a generic integrable quantum system coincide with those of a poisson process we prove that in the case of a rectangle billiard with random ratio of sides the sum behaves for random and n large like a random walk in the complex plane with a non gaussian limit distribution the expectation value of the distribution is zero its variance which is essentially the average pair correlation function is one in accordance with the berry tabor conjecture but all higher moments 4 diverge the proof of the existence of the limit distribution uses the mixing property of a dynamical system defined on a product of hyperbolic surfaces the berry tabor conjecture and the existence of the limit distribution for a fixed generic rectangle are related to an equidistribution conjecture for long horocycles on this product space

gaps in n mod 1 and ergodic theory cut the unit circle s r z at the points 1 2 n where x x mod 1 and let j1 jn denote the complementary intervals or gaps that remain we show that in contrast to the case of random points whose gaps are exponentially distributed the lengths ji n are governed by an explicit piecewise real analytic distribution f t dt with phase transitions at t 1 2 and t 2 the gap distribution is related to the probability p t that a random unimodular lattice translate r meets a fixed triangle st of area t in fact p t f t the proof uses ergodic theory on the universal elliptic curve e sl2 r r 2 sl2 z z and ratner s theorem on unipotent invariant measures

on the uniform equidistribution of closed horospheres in hyperbolic manifolds we prove equidistribution results for pieces of closed horospheres in cofinite hyperbolic manifolds of dimension n 1 using spectral methods this extends earlier results by hejhal hej2 and str

zimmer s conjecture for actions of mathrm sl m mathbb z we prove zimmer s conjecture for c 2 actions by finite index subgroups of mathrm sl m mathbb z provided m 3 the method utilizes many ingredients from our earlier proof of the conjecture for actions by cocompact lattices in mathrm sl m mathbb r brown et al in zimmer s conjecture subexponential growth measure rigidity and strong property t 2016 arxiv 1608 04995 but new ideas are needed to overcome the lack of compactness of the space g times m gamma admitting the induced g action non compactness allows both measures and lyapunov exponents to escape to infinity under averaging and a number of algebraic geometric and dynamical tools are used control this escape new ideas are provided by the work of lubotzky mozes and raghunathan on the structure of nonuniform lattices and in particular of mathrm sl m mathbb z providing a geometric decomposition of the cusp into rank one directions whose geometry is more easily controlled the proof also makes use of a precise quantitative form of non divergence of unipotent orbits by kleinbock and margulis and an extension by de la salle of strong property t to representations of nonuniform lattices

rational points of bounded height on fano varieties a prime pe7z let v be an algebraic variety defined over f and li a metrized line bundle on v i e a system l consisting of a line bundle l and a family of banach v adic metrics on l f for all v s satisfying well known conditions cf 1 chap vi then we can define a height function hl x f i r by the formula h x is x 1 where s is a local section of l non vanishing

decay of matrix coefficients on reductive homogeneous spaces of spherical type let z be a homogeneous space z g h of a real reductive lie group g with a reductive subgroup h the investigation concerns the quantitative decay of matrix coefficients on z under the assumption that z is of spherical type that is minimal parabolic subgroups have open orbits on z

the behavior under projection of dilating sets in a covering space let m be a compact riemannian manifold with covering space s and suppose dcur o r oo is a family of borel probability measures on s all of which arise from some fixed measure by r homotheties of s about some point followed by renormalization of the resulting measure in this paper we study the ergodic properties as a function of r of the corresponding family of projected measures on m in the euclidean and hyperbolic cases a typical example arises by considering the behavior of a dilating family of spheres under projection suppose tn is the n dimensional integral torus regarded as the quotient of rn by the lattice r of integral translations and suppose d1i is a borel probability measure on rn i e d1i is a real nonnegative borel measure on rn having total mass 1 by projection d i gives rise to a borel probability measure dm on tn if we define the measure of a set in tn to be the measure with respect to d i of its total preimage in rn with respect to the projection map from rn to tn for various types of f invariant functions f on rn or what is the same thing functions on tn it is natural to study the difference 1 fn f x dx f x dm x tn tn where dx is normalized lebesgue measure this quantity arises for example in the study of equidistribution of sequences and in the study of ergodic properties of irrational lines in tn in order to study 1 suppose for the moment that f is coo with fourier series e cn e27i n x and suppose d y f e20i xy dc4 x is the fourier transform of d u then it is easily checked that the nth fourier coefficient of dm x is d n and since f is smooth the parseval equality can be applied to conclude that j f x dm x cn t n co ezcn n tn where the prime indicates that the origin is omitted from the summation but co ftn f x dx so 2 f x dx f x dm x s cn n tn tn i e from our point of view the study of 1 is equivalent to the study of the asymptotic behavior of it is the purpose of this paper to illustrate this principle received by the editors october 21 1983 1980 mathmatics suject czassification primary 22d40 x 1984 american mathematical society 0002 9947 84 1 00 25 per page 855 this content downloaded from 207 46 13 149 on mon 03 oct 2016 06 26 09 utc all use subject to http about jstor org terms

lattice point asymptotics and volume growth on teichmuller space we apply some of the ideas of the ph d thesis of g a margulis mar70 to teichmuller space let x be a point in teichmuller space and let br x be the ball of radius r centered at x with distances measured in the teichmuller metric we obtain asymptotic formulas as r tends to infinity for the volume of br x and also for the cardinality of the intersection of br x with an orbit of the mapping class group

limiting distributions of curves under geodesic flow on hyperbolic manifolds we consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic n manifold under the geodesic flow suppose that the curve is not contained in a stable leaf of the flow it is shown that under the geodesic flow the normalized parameter measure on the curve gets asymptotically equidistributed with respect to the normalized natural riemannian measure on the unit tangent bundle of a closed totally geodesically immersed submanifold moreover if this immersed submanifold is a proper subset then a lift of the curve to the universal covering space t 1 h n is mapped into a proper subsphere of the ideal boundary sphere h n under the visual map this proper subsphere can be realized as the ideal boundary of an isometrically embedded hyperbolic subspace in h n covering the closed immersed submanifold in particular if the visual map does not send a lift of the curve into a proper subsphere of h n then under the geodesic flow the curve gets asymptotically equidistributed on the unit tangent bun dle of the manifold with respect to the normalized natural rie mannian measure the proof uses dynamical properties of unipotent flows on finite volume homogeneous spaces of so n 1

harmonic analysis and pointwise ergodic theorems for noncommuting transformations let fk denote the free group on k generators 1 where s n w i n and 1 denotes the word length on fk induced by s let x m m be a probability space on which fk acts ergodically by measure preserving transformations we prove a pointwise ergodic theorem for the sequence of operators ij i a a n acting on l 2 x namely a f x fx fdm almost everywhere for each f in l 2 x we also show that the sequence a2n converges to a conditional expectation operator with re spect to a a algebra which is invariant under fk the proof is based on the spectral theory of the commutative convolution subalgebra of e i fk gener ated by the elements an n 0 we then generalize the discussion to algebras arising as a gelfand pair associated with the group of automorphisms g ri r2 of a semi homogeneous tree t r r2 where r 2 r2 2 r r2 4 this content downloaded from 207 46 13 114 on thu 26 may 2016 06 27 50 utc all use subject to http about jstor org terms

ergodicity of generalised patterson sullivan measures in higher rank symmetric spaces let x g k be a higher rank symmetric space of noncompact type and open image in new window a discrete zariski dense group in a previous article we constructed for each g invariant subset of the regular limit set of a family of measures the so called b densities our main result here states that these densities are ergodic with respect to an important subset of the limit set which we choose to call the ray limit set in the particular case of uniform lattices and products of convex cocompact groups acting on the product of rank one symmetric spaces every limit point belongs to the ray limit set hence our result is most powerful for these examples for nonuniform lattices however it is a priori not clear whether the ray limit set has positive measure with respect to a b density using a counting theorem of eskin and mcmullen we are able to prove that the ray limit set has full measure in each g invariant subset of the limit set