fractal dimension of higher dimensional chaotic repellors abstract using examples we test formulae previously conjectured to give the fractal information dimension of chaotic repellors and their stable and unstable manifolds in typical dynamical systems in terms of the lyapunov exponents and the characteristic escape time from the repellor our main example a three dimensional chaotic scattering billiard yields a new structure for its invariant manifolds this system also provides an example of a system which is not typical and illustrates how perturbation to the system restores typicality and the applicability of the dimension formulae

new criteria for ergodicity and nonuniform hyperbolicity in this work we obtain a new criterion to establish ergodicity and nonuniform hyperbolicity of smooth measures of diffeomorphisms of closed connected riemannian manifolds this method allows us to give a more accurate description of certain ergodic components the use of this criterion in combination with topological devices such as blenders lets us obtain global ergodicity and abundance of nonzero lyapunov exponents in some contexts in the partial hyperbolicity context we obtain that stably ergodic diffeomorphisms are c1 dense among volume preserving partially hyperbolic diffeomorphisms with 2 dimensional center bundle this is motivated by a well known conjecture of pugh and shub

the metric entropy of endomorphisms we prove that for a c 2 non invertible but non degenerate map on a compact riemannian manifold without boundary an invariant measure satisfies an equality relating entropy folding entropy and negative lyapunov exponents this generalizes ledrappier young s entropy formula 5 for negative lyapunov exponents of diffeomorphisms to the case of endomorphisms

singular hyperbolic attractors are chaotic we prove that a singular hyperbolic attractor of a 3 dimensional flow is chaotic in two different strong senses first the flow is expansive if two points remain close at all times possibly with time reparametrization then their orbits coincide second there exists a physical or sinai ruelle bowen measure supported on the attractor whose ergodic basin covers a full lebesgue volume measure subset of the topological basin of attraction moreover this measure has absolutely continuous conditional measures along the center unstable direction is a u gibbs state and is an equilibrium state for the logarithm of the jacobian of the time one map of the flow along the strong unstable direction this extends to the class of singular hyperbolic attractors the main elements of the ergodic theory of uniformly hyperbolic or axiom a attractors for flows in particular these results can be applied i to the flow defined by the lorenz equations ii to the geometric lorenz flows iii to the attractors appearing in the unfolding of certain resonant double homoclinic loops iv in the unfolding of certain singular cycles and v in some geometrical models which are singular hyperbolic but of a different topological type from the geometric lorenz models in all these cases the results show that these attractors are expansive and have physical measures which are u gibbs states

replicator dynamics with turnover of players we study adaptive dynamics in games where players abandon the population at a given rate and are replaced by naive players characterized by a prior distribution over the admitted strategies we demonstrate how such a process leads macroscopically to a variant of the replicator equation with an additional term accounting for player turnover we study how nash equilibria and the dynamics of the system are modified by this additional term for prototypical examples such as the rock paper scissors game and different classes of two action games played between two distinct populations we conclude by showing how player turnover can account for nontrivial departures from nash equilibria observed in data from lowest unique bid auctions

diffeomorphisms approximated by anosov on the 2 torus and their sbr measures we consider the c2 set of c2 diffeomorphisms of the 2 torus t2 provided the conditions that the tangent bundle splits into the directed sum tt2 es ff eu of df invariant subbundles es eu and there is 0 1 then we prove that the set is the union of anosov diffeoinorphisms and diffeomorphisms approximated by anosov and moreover every diffeomorphism approximated by anosov in the c2 set has no sbr measures this is related to a result of hu young we know that anosov diffeomorphisms are structurally stable anosov a robbin ri and robinson r2 and have an sbr measure sinai si recently huyoung h y showed that a special diffeomorphism g of the 2 torus t2 provided the condition that for x e t 2 there are 0 1 and if p is the origin then iidgiei 11 1 has no sbr measures such a diffeomorphism is called almost anosov let diff2 t2 be a set of c2 diffeomorphisms on the 2 torus imposed with the c2 topology a diffeomorphism f t 2 t2is called anosov if f has a hyperbolic structure on all of t 2 cf s2 we denote as a t 2 the open set of anosov diffeomorphisms then a t 2 is a proper subset of the set 02 of diffeomorphisms such that iii the tangent bundle tt 2 splits into the directed sum tt 2 eu 33 es of invariant subbundles eu and es and iv there exist a c riemannian metric and 0 1 our aim is to investigate the dynamical properties of diffeomorphisms belonging to 02 a t 2 more precisely we state them as follows theorem a each diffeomorphism belonging to 02 a t 2 is approximated by anosov diffeomorphisms and it has no sbr measures the conclusions will be obtained in proving the following three propositions proposition b let f e 02 a t 2 then the set a defined by a x e t 2 12 idfnleu 11j 1 for n e z has the following properties received by the editors february 10 1997 1991 mathematics subject classification primary 58f11 58f12 58f15

entropy formula for random transformations we exhibit random strange attractors with random sinai bowen ruelle measures for the composition of independent random diffeomorphisms

a grobman hartman theorem for general nonuniform exponential dichotomies for a nonautonomous dynamics with discrete time given by a sequence of linear operators am we establish a version of the grobman hartman theorem in banach spaces for a very general nonuniformly hyperbolic dynamics more precisely we consider a sequence of linear operators whose products exhibit stable and unstable behaviors with respect to arbitrary growth rates ec n determined by a sequence n for all sufficiently small lipschitz perturbations am fm we construct topological conjugacies between the dynamics defined by this sequence and the dynamics defined by the operators am we also show that all conjugacies are holder continuous we note that the usual exponential behavior is included as a very special case when n n but many other asymptotic behaviors are included such as the polynomial asymptotic behavior when n logn

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

equilibrium states for factor maps between subshifts abstract let x y be a factor map where x x and y y are subshifts over finite alphabets assume that x satisfies weak specification let a a 1 a 2 r 2 with a 1 0 and a 2 0 let f be a continuous function on x with sufficient regularity holder continuity for instance we show that there is a unique shift invariant measure on x that maximizes f d a 1 h x a 2 h 1 y in particular taking f 0 we see that there is a unique invariant measure on x that maximizes the weighted entropy a 1 h x a 2 h 1 y which answers an open question raised by gatzouras and peres 1996 in 15 an extension is given to high dimensional cases as an application we show that for each compact invariant set k on the k torus under a diagonal endomorphism if the symbolic coding of k satisfies weak specification then there is a unique invariant measure supported on k so that dim h dim h k

dimension and product structure of hyperbolic measures we prove that every hyperbolic measure invariant under a c 1 fi difieomorphism of a smooth riemannian manifold possesses asymptotically almost local product structure i e its density can be approximated by the product of the densities on stable and unstable manifolds up to small exponentials this has not been known even for measures supported on locally maximal hyperbolic sets using this property of hyperbolic measures we prove the long standing eckmann ruelle conjecture in dimension theory of smooth dynamical systems the pointwise dimension of every hyperbolic measure invariant under a c 1 fi difieomorphism exists almost everywhere this implies the crucial fact that virtually all the characteristics of dimension type of the measure including the hausdorfi dimension box dimension and information dimension coincide this provides the rigorous mathematical justiflcation of the concept of fractal dimension for hyperbolic measures

regularity of srb entropy for geometric lorenz attractors we consider the classical geometric lorenz attractors showing that the srb entropy admits gamma holder continuity for any 0 gamma 1

universal behaviour of extreme value statistics for selected observables of dynamical systems the main results of the extreme value theory developed for the investigation of the observables of dynamical systems rely up to now on the block maxima approach in this framework extremes are identified with the block maxima of the time series of the chosen observable in the limit of infinitely long blocks it has been proved that assuming suitable mixing conditions for the underlying dynamical systems the extremes of a specific class of observables are distributed according to the so called generalised extreme value gev distribution direct calculations show that in the case of quasi periodic dynamics the block maxima are not distributed according to the gev distribution in this paper we show that considering the exceedances over a given threshold instead of the block maxima approach it is possible to obtain a generalised pareto distribution also for extremes computed in systems which do not satisfy mixing conditions requiring that the invariant measure locally scales with a well defined exponent the local dimension we show that the limiting distribution for the exceedances of the observables previously studied with the block maxima approach is a generalised pareto distribution where the parameters depend only on the local dimensions and the values of the threshold but not on the number of observations considered we also provide connections with the results obtained with the block maxima approach in order to provide further support to our findings we present the results of numerical experiments carried out considering the well known chirikov standard map

entropy of semiclassical measures in dimension 2 we study the high energy asymptotic properties of eigenfunctions of the laplacian in the case of a compact riemannian surfacem of anosov type to do this we look at families of distributions associated to them on the cotangent bundle t m and we derive entropic properties on their accumulation points in the high energy limit the so called semiclassical measures we show that the kolmogorov sinai entropy of a semiclassical measure for the geodesic ow gt is bounded from below by half of the ruelle upper bound i e hks g 1 2 s m d where is the upper lyapunov exponent at point

hausdorff dimension of harmonic measures on negatively curved manifolds we show by probabilistic means that harmonic measures on manifolds whose curvature is sandwiched between two negative constants have positive hausdorff dimensions a lower bound for harmonic measures of open sets is derived as well we end with the results concerning the hausdorff dimension of harmonic measures on universal covers of compact negatively curved manifolds

locating resonances for axiom a dynamical systems for a class of differentiable dynamical systems called axiom a systems it has been shown by pollicott and the author that correlation functions have fourier transforms which are meromorphic in a strip the poles or resonances are however not easy to locate this note reviews the results which are known and discusses a simple model where the position of resonances can be estimated the effect of noise is also discussed

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

invariant measures and arithmetic quantum unique ergodicity we classify measures on the locally homogeneous space i sl 2 r l which are invariant and have positive entropy under the diagonal subgroup of sl 2 r and recurrent under l this classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces and a similar but slightly weaker result for the finite volume case other applications are also presented in the appendix joint with d rudolph we present a maximal ergodic theorem related to a theorem of hurewicz which is used in theproof of the main result

smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics this paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics we adopt a new point of view which has emerged progressively in recent years and which takes seriously into account the chaotic character of the microscopic time evolution the emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of boltzmann the nonequilibrium steady states in presence of a gaussian thermostat are described by srb measures in terms of these one can prove the gallavotti cohen fluctuation theorem one can also prove a general linear response formula and study its consequences which are not restricted to near equilibrium situations at equilibrium one recovers in particular the onsager reciprocity relations under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of dettmann and morriss the results just mentioned hold so far only for classical systems they do not involve large size i e they hold without a thermodynamic limit

ledrappier young formula and exact dimensionality of self affine measures in this paper we solve the long standing open problem on exact dimensionality of self affine measures on the plane we show that every self affine measure on the plane is exact dimensional regardless of the choice of the defining iterated function system in higher dimensions under certain assumptions we prove that self affine and quasi self affine measures are exact dimensional in both cases the measures satisfy the ledrappier young formula

ruelle inequality of folding type for c 1 alpha maps for all c 1 alpha maps we prove an inequality conjectured by ruelle that the metric entropy is bounded from above by the folding entropy minus the sum of the negative lyapunov exponents

generalizations of srb measures to nonautonomous random and infinite dimensional systems we review some developments that are direct outgrowths of or closely related to the idea of srb measures as introduced by sinai ruelle and bowen in the 1970s these new directions of research include the emergence of strange attractors in periodically forced dynamical systems random attractors in systems defined by stochastic differential equations srb measures for infinite dimensional systems including those defined by large classes of dissipative pdes quasi static distributions for slowly varying time dependent systems and surviving distributions in leaky dynamical systems

entropy chaos and weak horseshoe for infinite dimensional random dynamical systems in this paper we study the complicated dynamics of infinite dimensional random dynamical systems that include deterministic dynamical systems as their special cases in a polish space without assuming any hyperbolicity we prove if a continuous random map has a positive topological entropy then it contains a topological horseshoe we also show that the positive topological entropy implies the chaos in the sense of li yorke the complicated behavior exhibited here is induced by the positive entropy but not the randomness of the system 2017 wiley periodicals inc

empirical measures of partially hyperbolic attractors in this paper we study the limit measures of the empirical measures of lebesgue almost every point in the basin of a partially hyperbolic attractor they are strongly related to a notion named gibbs u state which can be defined in a large class of diffeomorphisms with less regularity and which is the same as pesin sinai s notion for partially hyperbolic attractors of c 1 alpha diffeomorphisms in particular we prove that for partially hyperbolic c 1 alpha diffeomorphisms with one dimensional center and for lebesgue almost every point 1 the center lyapunov exponent is well defined but 2 the sequence of empirical measures may not converge in order to prove 2 we build a diffeomorphism with historical behavior which is transitive contrary to the well known example of bowen we also give some consequences on srb measures and large deviations

nonuniform measure rigidity we consider an ergodic invariant measure for a smooth actionof z k k 2 on a k 1 dimensional manifold or for a locally free smooth action of r k k 2 on a 2k 1 dimensional manifold we prove that if is hyperbolic with the lyapunov hyperplanes in general position and if one element in z k has positive entropy then is absolutely continuous the main ingredient is absolute continuity of conditional measures on lyapunov foliations which holds for a more general class of smooth actions of higher rank abelian groups

the metric entropy of diffeomorphisms part i characterization of measures satisfying pesin s entropy formula soit m une variete de riemann compacte soit f m m un diffeomorphisme et soit m une mesure de probabilite de borel f invariante sur m on identifie les mesures pour lesquelles l inegalite de margulis et ruelle qui relie l entropie aux exposants de lyapunov atteint l egalite on demontre une formule valable pour toutes les mesures invariantes

sinai bowen ruelle measures for certain henon maps we study maps t a b 2 2 defined by t a b left x y right left 1 a x 2 y bx right 0 0

local product structure for equilibrium states the usual way to study the local structure of equilibrium state of an axioma diffeomorphism or flow is to use the symbolic dynamic and to push results on the manifold a new geometrical method is given it consists in proving that equilibrium states for holder continuous functions are related to other equilibrium states of some special sub systems satisfying a sort of expansiveness using different kinds of extensions the local product structure of gibbs measure is proven

non stationary non uniform hyperbolicity srb measures for dissipative maps we prove the existence of srb measures for diffeomorphisms where a positive volume set of initial conditions satisfy an effective hyperbolicity condition that guarantees certain recurrence conditions on the iterates of lebesgue measure we give examples of systems that do not admit a dominated splitting but can be shown to have srb measures using our methods

srb measures for attractors with continuous invariant splittings we study the existence of srb measures of c 2 diffeomorphisms for attractors whose bundles admit holder continuous invariant non dominated splittings we prove the existence when one sub bundle has the non uniform expanding property on a set with positive lebesgue measure and the other sub bundle admits non positive lyapunov exponents on a total probability set

invariant measures satisfying an equality relating entropy folding entropy and negative lyapunov exponents in this paper we prove that for a c2 non invertible but non degenerate map on a compact manifold an invariant measure satisfies an equality relating entropy folding entropy and negative lyapunov exponents if and under a condition on the jacobian of the map only if the measure has absolutely continuous conditional measures on the stable manifolds

srb measures and pesin s entropy formula for endomorphisms a celebrated result conjectured by ruelle and proved by ledrappier strelcyn and young in the smooth ergodic theory of diffeomorphisms is that an invariant measure of a c 2 diffeomorphism satisfies pesin s entropy formula if and only if it has absolutely continuous conditional measures on unstable manifolds 41 42 the later property is known as srb property of the invariant measure this result was also successfully generalized to random dynamical systems of diffeomorphisms 51

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

preimage pressure stable pressure and equilibrium states abstract in this paper the pressure of a potential via the dynamics of preimage structure are investigated for noninvertible systems and the properties for the related equilibrium states are considered for a continuous map on a compact metric space a notion of preimage pressure via the preimages of single points is introduced and then some fundamental properties are considered and a variational principle is established for the system with uniform separation of preimages for a c 1 partially hyperbolic endomorphism on a closed riemannian manifold a notion of stable pressure via the preimages of local stable manifolds is introduced and a corresponding variational principle is established moreover equilibrium states and tangent functionals for pointwise preimage pressures and stable pressures are also considered by describing the set of equilibrium states and tangent functionals and a sense of continuous dependence of equilibrium states with respect to the potential we prove that the continuous potentials with unique equilibrium state form a dense g set in addition for any finite collection of ergodic measures we can find a continuous potential such that its set of equilibrium states contains the given set

quantitative boltzmann gibbs principles via orthogonal polynomial duality we study fluctuation fields of orthogonal polynomials in the context of particle systems with duality we thereby obtain a systematic orthogonal decomposition of the fluctuation fields of local functions where the order of every term can be quantified this implies a quantitative generalization of the boltzmann gibbs principle in the context of independent random walkers we complete this program including also fluctuation fields in non stationary context local equilibrium for other interacting particle systems with duality such as the symmetric exclusion process similar results can be obtained under precise conditions on the n particle dynamics

rate of escape from nonattracting chaotic sets chaotic transient phenomena occur in the vicinity of nonattracting chaotic sets the rate of escape measures the average length of the transients there is a conjecture by eckmann and ruelle connecting the rate of escape to the lyapunov exponents and entropy we prove an inequality that partially supports the conjecture

dimensions of attractors in pinched skew products we study dimensions of strange non chaotic attractors and their associated physical measures in so called pinched skew products introduced by grebogi and his coworkers in 1984 our main results are that the hausdorff dimension the pointwise dimension and the information dimension are all equal to one although the box counting dimension is known to be two the assertion concerning the pointwise dimension is deduced from the stronger result that the physical measure is rectifiable our findings confirm a conjecture by ding grebogi and ott from 1989

entropy and the localization of eigenfunctions we study the large eigenvalue limit for the eigenfunctions of the laplacian on a compact manifold of negative curvature in fact we only assume that the geodesic flow has the anosov property in the semi classical limit we prove that the wigner measures associated to eigenfunctions have positive metric entropy in particular they cannot concentrate entirely on closed geodesies 1 introduction statement of results we consider a compact riemannian manifold m of dimension d 2 and assume that the geodesic flow ter acting on the unit tangent bundle of m has a chaotic behaviour this refers to the asymptotic properties of the flow when time t tends to infinity ergodicity mixing hyperbolicity we assume here that the geodesic flow has the anosov property the main example being the case of negatively curved manifolds the words quantum chaos express the intuitive idea that the chaotic features of the geodesic flow should imply certain special features for the corresponding quantum dynamical system that is according to schrodinger the unitary flow exp iht t r acting on the hilbert space l2 m where a stands for the laplacian on m and h is proportional to the planck constant recall that the quantum flow converges in a sense to the classical flow g in the so called semi classical limit h 0 one can imagine that for small values ofh the quantum system will inherit certain qualitative properties of the classical flow one expects for instance a very different behaviour of eigenfunctions of the laplacian or the distribution of its eigenvalues if the geodesic flow is anosov or in the other extreme completely integrable see sa95 the convergence of the quantum flow to the classical flow is stated in the egorov theorem consider one of the usual quantization procedures op which associates an operator oph a acting on l2 m to every smooth compactly supported function a e c t m on the cotangent bundle t m according to the egorov theorem we have for any fixed t

fluctuations of lyapunov vectors in large volume limit and the information flow configuration in space time dynamics abstract the fluctuation of lyapunov vectors in r n for large n and in the infinite dimensional limit are studied in the context of entropy balance from dynamical system theory the density of the fluctuation yields the information flow configuration ifc which captures the interaction of degrees of freedom dof in information flow terms generally the ifc can be extended or localized the study of these cases leads to the local formulation of entropy balance for a dof two examples of coupled map lattice are given to demonstrate the approach based on the numerical data we conjecture that a locally balance imbalance configuration of the extended localized ifc implies absence presence of pattern in the spatially extended system moreover for dissipative dynamics the localization for the largest and smallest exponents can take place in a fixed spatial domain which describes a stronger version fluctuation dissipation principle derived by ruelle 1982 in viscous turbulence flow some of its implications are discussed

entropy formula of pesin type for noninvertible random dynamical systems in this paper we consider random dynamical systems abbreviated henceforth as rds s generated by compositions of random endomorphisms maybe noninvertible and with singularities of class c 2 of a compact manifold entropy formula of pesin type is proved for such rds s under some absolute continuity conditions on the associated invariant measures

the metric entropy of diffeomorphisms soit m une variete de riemann compacte c sans bord soit f un diffeomorphisme c 2 de m et soit m une mesure de probabilite de borel f invariante sur m soit h m f l entropie de f on donne des relations entre h m f et les proprietes geometriques de f et de m

ruelle inequality relating entropy folding entropy and negative lyapunov exponents in this paper we prove an inequality conjectured by ruelle relating the entropy folding entropy and negative lyapunov exponents of a differentiable map on a compact manifold under a set of conditions on degenerate points of the map

a derivation of the broadwell equation we consider a stochastic system of particles in a two dimensional lattice and prove that under a suitable limit i e n e 0 ne2 const wheren is the number of particles and e is the mesh of the lattice the one particle distribution function converges to a solution of the two dimensional broadwell equation for all times for which the solution of this equation exists propagation of chaos is also proven

the work of einsiedler katok and lindenstrauss on the littlewood conjecture this document is intended as a slightly expanded writeup of my anticipated talk at the ams current events bulletin in new orleans january 2007 it is a brief report on the work of einsiedler katok and lindenstrauss on the littlewood conjecture 5 it is not intended in any sense for specialists and is indeed aimed at readers without any specific background either in measure theory dynamics or number theory any reader with any background in ergodic theory will be better served by consulting either the original paper or one of the surveys written by those authors see 7 and 12

entropy volume growth and srb measures for banach space mappings we consider c 2 frechet differentiable mappings of banach spaces leaving invariant compactly supported borel probability measures and study the relation between entropy and volume growth for a natural notion of volume defined on finite dimensional subspaces srb measures are characterized as exactly those measures for which entropy is equal to volume growth on unstable manifolds equivalently the sum of positive lyapunov exponents of the map in addition to numerous difficulties incurred by our infinite dimensional setting a crucial aspect to the proof is the technical point that the volume elements induced on unstable manifolds are regular enough to permit distortion control of iterated determinant functions the results here generalize previously known results for diffeomorphisms of finite dimensional riemannian manifolds and are applicable to dynamical systems defined by large classes of dissipative parabolic pdes

physical measures for multivalued inverse iterates near hyperbolic repellors we study new invariant probability measures describing the distribution of mul tivalued inverse iterates i e of different local inverse iterates for a non invertible smooth function f which is hyperbolic but not necessarily expanding on a repellor the methods for the higher dimensional non expanding and non invertible case are different than the ones for diffeomorphisms due to the lack of a nice unstable foliation local unstable manifolds depend on prehistories and may intersect each other both inand outside and the fact that markov partitions may not exist on we obtain that for lebesgue almost all points z in a neighbourhood v of the normalized averages of dirac measures on the consecu tive preimage sets of z converge weakly to an equilibrium measure on this implies that is a physical measure for the local inverse iterates of f it turns out that is an inverse srb measure in the sense that it is the only invariant measure satisfying a pesin type formula for the negative lyapunov exponents also we show that has absolutely continuous conditional measures on local stable manifolds by using the above convergence of measures we prove then that f b b cannot be one sided bernoulli although it is an exact endomorphism of lebesgue spaces several classes of ex amples of hyperbolic non invertible and non expanding repellors with their inverse srb measures are given in the end

entropy of physical measures for c infty c dynamical systems for a c infty map f on a compact manifold m we prove that for a lebesgue randomly picked point x there is an empirical measure from x with entropy larger than or equal to the top lyapunov exponent of lambda df lambda tm circlearrowleft at x this contrasts with the well known ruelle inequality as a consequence we give some refinement of tsujii s work 23 relating physical and sinai ruelle bowen measures

dimensions of invariant sets of expanding maps we consider a compact invariant set of an expanding map of a manifoldm and give upper and lowerbounds for the hausdorff dimension dimh and box dimensionsdimb and dimb these bounds are given in terms of the topological entropy topological pressure and uniform lyapunov exponents of the map

conformality for a robust class of non conformal attractors in this paper we investigate the hausdorff dimension of limit sets of anosov representations in this context we revisit and extend the framework of hyperconvex representations and establish a convergence property for them analogue to a differentiability property as an application of this convergence we prove that the hausdorff dimension of the limit set of a hyperconvex representation is equal to a suitably chosen critical exponent in the appendix in collaboration with m bridgeman we extend a classical result on the hessian of the hausdorff dimension on purely imaginary directions

dimension theory for invariant measures of endomorphisms we establish the exact dimensional property of an ergodic hyperbolic measure for a c 2 non invertible but non degenerate endomorphism on a compact riemannian manifold without boundary based on this we give a new formula of lyapunov dimension of ergodic measures and show it coincides with the dimension of hyperbolic ergodic measures in a setting of random endomorphisms our results extend several well known theorems of barreira et al ann math 149 755 783 1999 and ledrappier and young commun math phys 117 4 529 548 1988 for diffeomorphisms to the case of endomorphisms

self organized criticality and thermodynamic formalism we develop a thermodynamic formalism for a dissipative version of the zhang model of self organized criticality where a parameter allows us to tune the local energy dissipation by constructing a suitable markov partition we define gibbs measures in the sense of sinai ruelle and bowen partition functions and topological pressure allowing the analysis of probability distributions of avalanches we discuss the infinite size limit in this setting in particular we show that a lee yang phenomenon occurs in the conservative case this suggests new connections to classical critical phenomena

statistical stability and continuity of srb entropy for systems with gibbs markov structures we present conditions on families of diffeomorphisms that guarantee statistical stability and srb entropy continuity they rely on the existence of horseshoe like sets with infinitely many branches and variable return times as an application we consider the family of henon maps within the set of benedicks carleson parameters

when a physical like measure is physical or srb abstract for partially hyperbolic systems we show that 1 integrable expanding along central bundle of all physical like measures imply the set of physical like is composed of finite ergodic hyperbolic srb measures 2 a physical like measure with integrable expanding along central bundle is an ergodic hyperbolic srb 3 quasi conformal central bundle implies the set of physical like measures is contained in the set of hyperbolic srb measures and in particular implies the existence of physical measures

unstable entropies and variational principle for partially hyperbolic diffeomorphisms abstract we study entropies caused by the unstable part of partially hyperbolic systems we define unstable metric entropy and unstable topological entropy and establish a variational principle for partially hyperbolic diffeomorphisms which states that the unstable topological entropy is the supremum of the unstable metric entropy taken over all invariant measures the unstable metric entropy for an invariant measure is defined as a conditional entropy along unstable manifolds and it turns out to be the same as that given by ledrappier young though we do not use increasing partitions the unstable topological entropy is defined equivalently via separated sets spanning sets and open covers along a piece of unstable leaf and it coincides with the unstable volume growth along unstable foliation we also obtain some properties for the unstable metric entropy such as affineness upper semi continuity and a version of shannon mcmillan breiman theorem

dimension of hyperbolic measures of random diffeomorphisms we consider dynamics of compositions of stationary random c 2 diffeomorphisms we will prove that the sample measures of an ergodic hyperbolic invariant measure of the system are exact dimensional this is an extension to random diffeomorphisms of the main result of barreira pesin and schmeling 1999 which proves the eckmann ruelle dimension conjecture for a deterministic diffeomorphism

dimension theory of iterated function systems let si l i 1 be an iterated function system ifs on d with attractor k let denote the one sided full shift over the alphabet 1 l we define the projection entropy function h on the space of invariant measures on e associated with the coding map k and develop some basic ergodic properties about it this concept turns out to be crucial in the study of dimensional properties of invariant measures on k we show that for any conformal ifs respectively the direct product of finitely many conformal ifss without any separation condition the projection of an ergodic measure under is always exactly dimensional and its hausdorff dimension can be represented as the ratio of its projection entropy to its lyapunov exponent respectively the linear combination of projection entropies associated with several coding maps furthermore for any conformal ifs and certain affine ifss we prove a variational principle between the hausdorff dimension of the attractors and that of projections of ergodic measures 2008 wiley periodicals inc

entropy production and information gain in axiom a systems anosov systems are mathematical models for chaotic systems in statistical mechanics and fluid dynamics most of these systems enjoy the property of positive entropy production we introduce the concept of specific information gain or specific relative entropy h in anosov systems and prove that it is identical to the entropy production rate e p defined by ruelle and gallavotti in anosov systems from this point of view the entropy production rate e p characterizes the degree of macroscopic irreversibility of the system

some large deviation results for dynamical systems we prove some large deviation estimates for continuous maps of compact metric spaces and apply them to attractors in differentiable dynamics rate of escape problems and to shift spaces

generalized dimensions entropies and liapunov exponents from the pressure function for strange sets for conformal mixing repellers such as julia sets and nonlinear one dimensional cantor sets we connect the pressure of a smooth transformation on the repeller with its generalized dimensions entropies and liapunov exponents computed with respect to a set of equilibrium gibbs measures this allows us to compute the pressure by means of simple numerical algorithms our results are then extended to axiom a attractors and to a nonhyperbolic invariant set of the line in this last case we show that a first order phase transition appears in the pressure

the geometric approach for constructing sinai ruelle bowen measures an important class of physically relevant measures for dynamical systems with hyperbolic behavior is given by sinai ruelle bowen srb measures we survey various techniques for constructing srb measures and studying their properties paying special attention to the geometric push forward approach after describing this approach in the uniformly hyperbolic setting we review recent work that extends it to non uniformly hyperbolic systems