on simultaneous linearization of diffeomorphisms of the sphere let r1 r2 rm be rotations generating sod 1 d 2 and f1 f2 fm be their small smooth perturbations we show that f can be simultaneously linearized if and only if the associated random walk has zero lyapunov exponents as a consequence we obtain stable ergodicity of actions of random rotations in even dimensions 1 main results let f1 f2 fm be diffeomorphisms of s d 2 let i i n be a sequence of independent random variables uniformly distributed on 1 m consider the markov process on s 1 xn f nxn 1 if is an invariant measure for this process let 1 2 d be the lyapunov exponents of denote

rigidity for anosov actions of higher rank lattices this is the simplest example of a large class of analytic standard actions of lattices in semisimple lie groups on locally homogeneous spaces a basic problem is to understand the differentiable actions near to such a standard action in terms of their geometry and dynamics cf 13 50 51 a cr action so if x x x of a group f on a compact manifold x is said to be anosov if there exists at least one element yh e f such that p yh is an anosov diffeomorphism of x we begin in this paper to study the anosov differentiable actions of lattices including many standard algebraic examples and especially to study their stability properties our main theme is that either the cr rigidity or the cr deformation rigidity of an anosov action for 1 r cr or even for the real analytic case can be shown just by studying the behavior of the periodic orbits for the action there are two notions of structural stability that appear in this paper rigidity and deformation rigidity a cl perturbation of a cr action so is simply another cr action spl such that for a finite set of generators 81 ad of f the cr diffeomorphisms p 8i and pl 8i are cl close for all i an action so is said to be cr rigid or topologically rigid if r 0 if every sufficiently small c perturbation of so is cr conjugate to so for 0 r oo or r w in the case of real analytic actions a cl deformation of an action so is a continuous path of cr actions pt defined for some 0 t e with spo sd an action sd is said to be cr deform

holomorphic differentials thermostats and anosov flows we introduce a new family of thermostat flows on the unit tangent bundle of an oriented riemannian two manifold suitably reparametrised these flows include the geodesic flow of metrics of negative gauss curvature and the geodesic flow induced by the hilbert metric on the quotient surface of divisible convex sets we show that the family of flows can be parametrised in terms of certain weighted holomorphic differentials and investigate their properties in particular we prove that they admit a dominated splitting and we identify special cases in which the flows are anosov in the latter case we study when they admit an invariant measure in the lebesgue class and the regularity of the weak foliations

on the period spectrum of a symplectic mapping let m be a compact riemannian manifold it has been known for a long time that the singularities of the wave trace trace cos t are located at the periods of the closed geodesics do these singularities also contain information about the geometry of m in the neighborhood of a closed geodesic we prove that the birkhoff canonical form of the poincare map can be determined from the singularities of the wave trace

instability of high dimensional hamiltonian systems multiple resonances do not impede diffusion abstract we consider models given by hamiltonians of the form h i p q t e h i j 1 n 1 2 p j 2 v j q j e q i p q t e where i i r d t d p q r n t t 1 these are higher dimensional analogues both in the center and hyperbolic directions of the models studied in 28 29 43 and are usually called a priori unstable hamiltonian systems all these models present the large gap problem we show that for 0 e 1 under regularity and explicit non degeneracy conditions on the model there are orbits whose action variables i perform rather arbitrary excursions in a domain of size o 1 this domain includes resonance lines and hence large gaps among d dimensional kam tori this phenomenon is known as arnold diffusion the method of proof follows closely the strategy of 28 29 the main new phenomenon that appears when the dimension d of the center directions is larger than one is the existence of multiple resonances in the space of actions i i r d we show that since these multiple resonances happen in sets of codimension greater than one in the space of actions i they can be contoured this corresponds to the mechanism called diffusion across resonances in the physics literature the present paper however differs substantially from 28 29 on the technical details of the proofs we have taken advantage of the theory of the scattering map developed in 31 notably the symplectic properties which were not available when the above papers were written we have analyzed the conditions imposed on the resonances in more detail more precisely we have found that there is a simple condition on the melnikov potential which allows us to conclude that the resonances are crossed in particular this condition does not depend on the resonances so that the results are new even when applied to the models in 28 29

dynamical spectral rigidity among mathbb z _2 symmetric strictly convex domains close to a circle appendix b coauthored with h hezari

asymptotic properties of monte carlo estimators of diffusion processes abstract this paper studies the limit distributions of monte carlo estimators of diffusion processes we examine two types of estimators based on the euler scheme one applied to the original processes the other to a doss transformation of the processes we show that the transformation increases the speed of convergence of the euler scheme we also study estimators of conditional expectations of diffusions after characterizing expected approximation errors we construct second order bias corrected estimators we also derive new convergence results for the mihlstein scheme illustrations of the results are provided in the context of simulation based estimation of diffusion processes

smooth conjugacy and s r b measures for uniformly and non uniformly hyperbolic systems we give a new proof of the fact that the eigenvalues at corresponding periodic orbits forms a complete set of invariants for the smooth conjugacy of low dimensional anosov systems we also show that if a homeomorphism conjugating two smooth low dimensional anosov systems is absolutely continuous then it is as smooth as the maps we furthermore prove generalizations of these facts for non uniformly hyperbolic systems as well as extensions and counterexamples in higher dimensions

formulas for the derivative and critical points of topological entropy for anosov and geodesic flows this paper represents part of a program to understand the behavior of topological entropy for anosov and geodesic flows in this paper we have two goals first we obtain some regularity results forc1 perturbations second and more importantly we obtain explicit formulas for the derivative of topological entropy these formulas allow us to characterize the critical points of topological entropy on the space of negatively curved metrics

invariant pre foliations for non resonant non uniformly hyperbolic systems given an orbit whose linearization has invariant subspaces satisfying some non resonance conditions in the exponential rates of growth we prove existence of invariant manifolds tangent to these subspaces the exponential rates of growth can be understood either in the sense of lyapunov exponents or in the sense of exponential dichotomies these manifolds can correspond to slow manifolds which characterize the asymptotic convergence let x i i n be a regular orbit of a c 2 dynamical system f let s be a subset of its lyapunov exponents assume that all the lyapunov exponents in s are negative and that the sums of lyapunov exponents in s do not agree with any lyapunov exponent in the complement of s denote by e s xi the linear spaces spanned by the spaces associated to the lyapunov exponents in s we show that there are smooth manifolds w s xi such that f w s xi w s xi 1 and t xi w s xi e s xi we establish the same results for orbits satisfying dichotomies and whose rates of growth satisfy similar non resonance conditions these systems of invariant manifolds are not in general a foliation

differentiability and analyticity of topological entropy for anosov and geodesic flows in this paper we investigate the regularity of the topological entropyh top forc k perturbations of anosov flows we show that the topological entropy varies almost as smoothly as the perturbation the results in this paper along with several related results have been announced in kkpw

invariant distributions and cohomology for geodesic flows and higher cohomology of higher rank anosov actions abstract we are motivated by a conjecture of a and s katok to study the smooth cohomologies of a family of weyl chamber flows the conjecture is a natural generalization of the livsic theorem to anosov actions by higher rank abelian groups it involves a description of top degree cohomology and a vanishing statement for lower degrees our main result proved in part ii verifies the conjecture in lower degrees for our systems and steps in the correct direction in top degree in part i we study our base case geodesic flows of finite volume hyperbolic manifolds we describe obstructions invariant distributions to solving the coboundary equation in unitary representations of the group of orientation preserving isometries of hyperbolic n space and we study sobolev regularity of solutions one byproduct is a smooth livsic theorem for geodesic flows of hyperbolic manifolds with cusps part i provides the tools needed in part ii for the main theorem

nonhomogeneous locally free actions of the affine group we classify locally free smooth actions of the real affine group on closed three dimensional manifolds up to smooth conjugacy in particular we show that there are non homogeneous actions when the manifold is not a rational homology sphere and its fundamental group is not solvable we say two right actions 1 and 2 of a lie group g on manifolds m1 and m2 are c conjugate if there exist an isomorphism of g and a c diffeomorphism h from m1 to m2 such that h 1 p g 2 h p g for any g 2 g and p 2 m1 the map h is called a c conjugacy between 1 and 2 we also say 1 and 2 are c orbit equivalent if there exists a c

the impact of surface higher order differentiability in two dimensional contact elements the aim of this work is to propose new contact elements of higher order of differentiability for analysing two dimensional frictionless contact problems several methods were proposed in the literature to solve the problem caused by the lack of continuity resulting from the discretization among them are bezier interpolation hermitian interpolation and splines one of the difficulties in using hermitian interpolation is to verify the partition of unity therefore new elements that satisfy the c 1 and c 2 continuity at the interface are presented in this paper these new contact elements are based on hermitian polynomials for ensuring a higher order of continuity the advantage is that this approach can be easily developed not only for linear elements but also for quadratic elements with higher order of differentiability the performance of different surface representations is assessed through a comparison with a c 0 surface discretization some numerical examples are used for assessing the accuracy and the convergence behaviour

stable manifolds to bounded solutions in possibly ill posed pdes abstract we prove several results establishing existence and regularity of stable manifolds for different classes of special solutions for evolution equations these equations may be ill posed a single specific solution an invariant torus filled with quasiperiodic orbits or more general manifolds of solutions in the later cases which include several orbits we also establish the invariant manifolds of an orbit depend smoothly on the orbit analytically in the case of quasi periodic orbits and finitely differentiably in the case of more general families we first establish a general abstract theorem which under suitable spectral non degeneracy analyticity assumptions on the linearized equation establishes the existence of the desired manifold related results appear in the literature but our results allow that the nonlinearity is unbounded and we obtain smoothness of the invariant manifolds this makes the results in this paper applicable to some several models of current interest that could not be treated otherwise we discuss in detail the boussinesq equation of water waves similar phenomena happen in other long wave approximations and complex ginzburg landau equation more recently we observed 11 that our results also apply to mean field games since the equations we consider may be ill posed part of the requirements for the stable manifold is that one can define the forward dynamics on them note also that the methods that are based in the existence of dynamics such as graph transform do not apply to ill posed equation we use the methods based on integral equations perron method associated with the partial dynamics but we need to take advantage of smoothing properties of the partial dynamics note that even if the families of solutions we started with are finite dimensional the stable manifolds may be infinite dimensional

on bundles that admit fiberwise hyperbolic dynamics this paper is devoted to rigidity of smooth bundles which are equipped with fiberwise geometric or dynamical structure we show that the fiberwise associated sphere bundle to a bundle whose leaves are equipped with continuously varying metrics of negative curvature is a topologically trivial bundle when either the base space is simply connected or more generally when the bundle is fiber homotopically trivial we present two very different proofs of this result a geometric proof and a dynamical proof we also establish a number of rigidity results for bundles which are equipped with fiberwise anosov dynamical systems finally we present several examples which show that our results are sharp in certain ways or illustrate necessity of various assumptions

kam tori and whiskered invariant tori for non autonomous systems abstract we consider non autonomous dynamical systems which converge to autonomous or periodic systems exponentially fast in time such systems appear naturally as models of many physical processes affected by external pulses we introduce definitions of non autonomous invariant tori and non autonomous whiskered tori and their invariant manifolds and we prove their persistence under small perturbations smooth dependence on parameters and several geometric properties if the systems are hamiltonian the tori are lagrangian manifolds we note that such definitions are problematic for general time dependent systems but we show that they are unambiguous for systems converging exponentially fast to autonomous the proof of persistence relies only on a standard implicit function theorem in banach spaces and it does not require that the rotations in the tori are diophantine nor that the systems we consider preserve any geometric structure we only require that the autonomous system preserves these objects in particular when the autonomous system is integrable we obtain the persistence of tori with rational rotational we also discuss fast and efficient algorithms for their computation the method also applies to infinite dimensional systems which define a good evolution e g pde s when the systems considered are hamiltonian we show that the time dependent invariant tori are isotropic hence the invariant tori of maximal dimension are lagrangian manifolds we also obtain that the un stable manifolds of whiskered tori are lagrangian manifolds we also include a comparison with the more global theory developed in blazevski and de la llave 2011

cohomology equations and commutators of germs of contact diffeomorphisms we study the group of germs of contact diffeomorphisms at a fixed point we prove that the abelianization of this group is isomorphic to the multiplicative group of real positive numbers the principal ingredient in this proof is a version of the sternberg linearization theorem in which the conjugating diffeomorphism preserves the contact structure

partial differentiability of invariant splittings a key feature of a general nonlinear partially hyperbolic dynamical system is the absence of differentiability of its invariant splitting in this paper we show that often partial derivatives of the splitting exist and the splitting depends smoothly on the dynamical system itself

statistical regularities of self intersection counts for geodesics on negatively curved surfaces let s be a compact surface with constant negative curvature 1 from among all closed geodesics on of length t choose one at random and letnt be the number of its self intersections we prove that for a certain constant 0 the random variable nt t t has a limit distribution as t we conjecture that for surfaces of variable negative curvature the order of magnitude of typical variations is t rather than t we also prove analogous results for generic geodesics that is geodesics whose initial tangent vectors are chosen randomly according to normalized liouville measure date november 10 2011 1991 mathematics subject classification primary 57m05 secondary 53c22 37d40

lyapunov exponents and rigidity of anosov automorphisms and skew products abstract in this paper we obtain local rigidity results for linear anosov diffeomorphisms in terms of lyapunov exponents more specifically we show that given an irreducible linear hyperbolic automorphism l with simple real eigenvalues with distinct absolute values any small perturbation preserving the volume and with the same lyapunov exponents is smoothly conjugate to l we also obtain rigidity results for skew products over anosov diffeomorphisms given a volume preserving partially hyperbolic skew product diffeomorphism f 0 over an anosov automorphism of the 2 torus we show that for any volume preserving perturbation f of f 0 with the same average stable and unstable lyapunov exponents the center foliation is smooth

livsic theorem for matrix cocycles we prove the liv sic theorem for arbitrary gl m r cocycles we con sider a hyperbolic dynamical system f x x and a holder continuous function a x gl m r we show that if a has trivial periodic data i e a f n 1 p a fp a p id for each periodic point p f n p then there exists a holder continuous function c x gl m r satisfying a x c fx c x 1 for all x2 x the main new ingredients in the proof are results of independent interest on relations between the periodic data lyapunov exponents and uniform estimates on growth of products along orbits for an arbitrary holder function a

regularity of functions smooth along foliations and elliptic regularity abstract we produce two sets of results arising in the analysis of the degree of smoothness of a function that is known to be smooth along the leaves of one or more foliations these foliations might arise from anosov systems and while each leaf is smooth the leaves might vary in a nonsmooth fashion one set of results gives microlocal regularity of such a function away from the conormal bundle of a foliation the other set of results gives local regularity of solutions to a class of elliptic systems with fairly rough coefficients such a regularity theory is motivated by one attack on the foliation regularity problem

entropy production in thermostats ii we show that an arbitrary anosov gaussian thermostat close to equilibrium has positive entropy poduction unless the external field e has a global potential the configuration space is allowed to have any dimension and magnetic forces are also allowed we also show the following non perturbative result suppose a gaussian thermostat satisfies k w sigma frac14 e sigma 2 0 for every 2 plane where kw is the sectional curvature of the associated weyl connection and e sigma is the orthogonal projection of e onto then the entropy production of any srb measure is positive unless e has a global potential a related non perturbative result is also obtained for certain generalized thermostats on surfaces

invariants for smooth conjugacy of hyperbolic dynamical systems ii we show that the eigenvalues of the derivatives at periodic points form a complete set of invariants for smooth local conjugacy of anosov diffeomorphisms oft2

the fried conjecture in small dimensions we study the twisted ruelle zeta function x s for smooth anosov vector fields x acting on flat vector bundles over smooth compact manifolds in dimension 3 we prove fried conjecture relating reidemeister torsion and x 0 in higher dimensions we show more generally that x 0 is locally constant with respect to the vector field x under a spectral condition as a consequence we also show fried conjecture for anosov flows near the geodesic flow on the unit tangent bundle of hyperbolic 3 manifolds this gives the first examples of non analytic anosov flows and geodesic flows in variable negative curvature where fried conjecture holds true

invariant distributions and time averages for horocycle flows there are infinitely many obstructions to existence of smooth solutions of the cohomological equation uu f where u is the vector field generating the horocycle flow on the unit tangent bundle sm of a riemann m surface of finite area and f is a given function on sm we study the sobolev regularity of these obstructions construct smooth solutions of the cohomological equation and derive asymptotics for the ergodic averages of horocycle flows

weil petersson metrics manhattan curves and hausdorff dimension in this note we want to relate the weil petersson metric on teichm uller space to the boundary correspondence between the actions on the boundary of fuchsian groups consider the space of riemann metrics g on a compact surface v with negative euler characteristic this can be endowed with a number of natural riemannian metrics of particular interest is the weil petersson metric whose definition was proposed by weil in 1958 based on earlier work of petersson cf 18 there is a particularly intuitive equivalent definition of the weil petersson metric using the second derivative of lengths of typical closed geodesics due to thurston and wolpert 19

time dependent scattering theory for odes and applications to reaction dynamics we develop a time dependent scattering theory for general vector fields in euclidean space we give conditions that ensure that the wave maps exist are smooth invertible and depend smoothly on parameters we then discuss the intertwining relations and how they can be used to compute stable unstable manifolds for time dependent normally hyperbolic invariant manifolds the theory is particularly effective for hamiltonian mechanics we also give perturbative calculations of the scattering map that are analogous to fermi s golden rule in quantum mechanics we apply this theory to a problem in transition state theory concerning the exposure of molecules to a laser pulse for a short time we present a method to compute invariant manifolds for the laser driven henon heilies system and give perturbative calculations of the change in the branching ratio

on deformation boundary rigidity and spectral rigidity of riemannian surfaces with no focal points although the main results of the paper are obtained in the two dimensional case the problems under consideration can be posed in an arbitrary dimension let m g be a compact riemannian manifold with boundary the boundary rigidity problem consists of determining the riemannian metric g on m up to isometries which are the identity on the boundary by measuring the geodesic distance dg between boundary points it is easy to see that the answer to this question is no in general we can find metrics g such that there exists x0 m satisfying dist x0 m supx y mdg x y then we can change the metric g in a neighborhood of x0 without changing the boundary distance function a compact riemannian manifold m g with boundary is called boundary rigid if the metric g is uniquely determined by dg up to isometries which are the identity on the boundary there are a few classes of metrics which are known to be boundary rigid see 13 for references in this paper we consider the linearized problem let g be a one parameter family of riemannian metrics on m such that dg dg0 e e for a geodesic a b m of the metric g with a b m it is easy to see that

on invariant volumes of codimension one anosov flows and the verjovsky conjecture we show that any topologically transitive codimension one anosov flow on a closed manifold is topologically equivalent to a smooth anosov flow that preserves a smooth volume by a classical theorem due to verjovsky any higher dimensional codimension one anosov flow is topologically transitive recently simic showed that any higher dimensional codimension one anosov flow that preserves a smooth volume is topologically equivalent to the suspension of an anosov diffeomorphism therefore our result gives a complete classification of codimension one anosov flows up to topological equivalence in higher dimensions

geometric properties of the scattering map of a normally hyperbolic invariant manifold given a normally hyperbolic invariant manifold for a map f whose sta ble and unstable invariant manifolds intersect transversally we consider its associated scattering map that is the map that given an asymptotic orbit in the past gives the asymptotic orbit in the future we show that when f and are symplectic resp exact symplectic then the scattering map is symplectic resp exact symplectic furthermore we show that in the exact symplectic case there are extremely easy formulas for the primitive function which have a variational interpretation as dierence of actions we use this geometric information to obtain ecien t perturbative calculations of the scattering map using deformation theory this perturbation theory generalizes and extends several results already obtained using the melnikov method analogous results are true for hamiltonian o ws the proofs are obtained by geometrically natural methods and do not involve the use of particular coordinate systems hence the results can be used to obtain intersection properties of objects of any type we also reexamine the calculation of the scattering map in a geodesic o w perturbed by a quasi periodic potential we show that the geometric theory reproduces the results obtained in dls06b using methods of fast slow systems moreover the geometric theory allows to compute perturbatively the dependence on the slow variables which does not seem to be accessible to the previous methods

analytic smoothing of geometric maps with applications to kam theory we show that finitely differentiable diffeomorphisms which are either symplectic volume preserving or contact can be approximated with analytic diffeomorphisms that are respectively symplectic volume preserving or contact we prove that the approximating functions are uniformly bounded on some complex domains and that the rate of convergence in cr norms of the approximation can be estimated in terms of the size of such complex domains and the order of differentiability of the approximated function as an application to this result we give a proof of the existence the local uniqueness and the bootstrap of regularity of kam tori for finitely differentiable symplectic maps the symplectic maps considered here are not assumed either to be written in action angle variables or to be perturbations of integrable systems our main assumption is the existence of a finitely differentiable parameterization of a maximal dimensional torus that satisfies a non degeneracy condition and that is approximately invariant the symplectic volume preserving and contact forms are assumed to be analytic

analyticity of the sinai ruelle bowen measure for a class of simple anosov flows we consider perturbations of the hamiltonian flow associated with the geodesic flow on a surface with constant negative curvature we prove that under a small perturbation not necessarily of hamiltonian character the sinai ruelle bowen measure associated with the flow exists and is analytic in the strength of the perturbation an explicit example of thermostated dissipative dynamics is considered

absolutely continuous convolutions of singular measures and an application to the square fibonacci hamiltonian author s damanik d gorodetski a solomyak b abstract 2015 we prove for the square fibonacci hamiltonian that the density of states measure is absolutely continuous for almost all pairs of small coupling constants this is obtained from a new result we establish about the absolute continuity of convolutions of measures arising in hyperbolic dynamics with exact dimensional measures

smooth dependence of thermodynamic limits of srb measures the thermodynamic limits of srb measures for uniformly hyperbolic sets are smoothly dependent on the map in an appropriate functional space

bootstrap for local rigidity of anosov automorphisms on the 3 torus we establish a strong form of local rigidity for hyperbolic automorphisms of the 3 torus with real spectrum namely let l mathbb t 3 to mathbb t 3 be a hyperbolic automorphism of the 3 torus with real spectrum and let f be a c 1 small perturbation of l then f is smoothly c infty conjugate to l if and only if obstructions to c 1 conjugacy given by the eigenvalues at periodic points of f vanish by combining our result and a local rigidity result of kalinin and sadovskaya ergod theory dyn syst 29 117 136 2009 for conformal automorphisms this completes the local rigidity program for hyperbolic automorphisms in dimension 3 our work extends de la llave marco moriyon 2 dimensional local rigidity theory commun math phys 109 368 378 1987 ergod theory dyn syst 17 3 649 662 1997 commun math phys 109 4 681 689 1987

invariant distributions beurling transforms and tensor tomography in higher dimensions in the recent articles paternain et al j differ geom 98 147 181 2014 invent math 193 229 247 2013 a number of tensor tomography results were proved on two dimensional manifolds the purpose of this paper is to extend some of these methods to manifolds of any dimension a central concept is the surjectivity of the adjoint of the geodesic ray transform or equivalently the existence of certain distributions that are invariant under the geodesic flow we prove that on any anosov manifold one can find invariant distributions with controlled first fourier coefficients the proof is based on subelliptic type estimates and a pestov identity we present an alternative construction valid on manifolds with nonpositive curvature based on the fact that a natural beurling transform on such manifolds turns out to be essentially a contraction finally we obtain uniqueness results in tensor tomography both on simple and anosov manifolds that improve earlier results by assuming a condition on the terminator value for a modified jacobi equation