on the discrete time version of the brussels formalism we develop a discrete time version of the so called brussels formalism in nonequilibrium statistical mechanics for continuous endomorphisms of a banach space we show that if the evolution operator u and the projector p are such that pu is a compact operator and the spectral radius of i p u i p is strictly less than the spectral radius of u then the formalism holds and the evolution operator is quasicompact

on the essential spectrum of the transfer operator for expanding markov maps the essential spectrum of the transfer operator for expanding markov maps of the interval is studied in detail to this end we construct explicityly an infinite set of eigenfunctions which allows us to prove that the essential spectrum inc k is a disk whose radius is related to the free energy of the liapunov exponent

from the hartree dynamics to the vlasov equation we consider the evolution of quasi free states describing n fermions in the mean field limit as governed by the nonlinear hartree equation in the limit of large n we study the convergence towards the classical vlasov equation for a class of regular interaction potentials we establish precise bounds on the 0rate of convergence

on the thermodynamic formalism for the gauss map we study the generalized transfer operator open image in new window of the gauss maptx 1 x mod 1 on the unit interval this operator which for 1 is the familiar perron frobenius operator oft can be defined for re 1 2 as a nuclear operator either on the banach spacea d of holomorphic functions over a certain discd or on the hilbert space open image in new window of functions belonging to some hardy class of functions over the half planeh 1 2 the spectra of open image in new window on the two spaces are identical on the space open image in new window is isomorphic to an integral operator open image in new window with kernel the bessel function mathfrak f 2 beta 1 2 sqrt st and hence to some generalized hankel transform this shows that open image in new window has real spectrum for real 1 2 on the spacea d the operator open image in new window can be analytically continued to the entire plane with simple poles at beta beta k 1 k 2 k 0 1 2 and residue the rank 1 operator mathcal n k f tfrac 1 2 1 k f k 0 from this similar analyticity properties for the fredholm determinant open image in new window of open image in new window and hence also for ruelle s zeta function follow another application is to the function zeta m beta sum limits n 1 infty n beta where n denotes the irrational n n n2 4 1 2 2 m extends to a meromorphic function in the plane with the only poles at 1 both with residue 1

on the kohn sham equations with periodic background potentials we study the question of existence and uniqueness for the finite temperature kohn sham equations for finite volumes a unique soluion is shown to exists if the effective potential satisfies a set of general conditions and the coupling constant is smaller than a certain value for periodic background potentials this value is proven to be volume independent in this case the finite volume solutions are shown to converge as the thermodynamic limit is considered the local density approximation is shown to satisfy the general conditions mentioned above

stability of statistical properties in two dimensional piecewise hyperbolic maps we investigate the statistical properties of a piecewise smooth dynamical system by studying directly the action of the transfer operator on appropriate spaces of distributions we accomplish such a program in the case of two dimensional maps with uniformly bounded second derivative but we are confident that the present approach can be successful in much greater generality we hope including higher dimensional billiards for the class of systems at hand we obtain a complete description of the srb measures their statistical properties and their stability with respect to many types of perturbations including deterministic and random perturbations and holes

on the fredholm determinant of a piecewise linear transformation abstract a piecewise linear transformation is realized on a signed symbolic dynamics by a renewal equation constructed on it a finite dimensional matrix z is defined the spectra of the perron frobenius operator associated with the transformation are characterized by the equation det i z 0 and the zeta function is also characterized by z

hartree approximation i the fixed point approach we consider the hartree approximation at finite temperature we give a justification of this approximation by using the methods of functional integration for finite temperatures a fixed point approach to solving the hartree problem is proposed for a class of two body interactions and background potentials that includes the coulomb interaction we prove that the hartree equation has a unique solution provided the coupling constant is small we also prove a similar result for local density theory

on the trace and the sum of elements of a matrix it is demonstrated that in many situations the sum of elements and the trace of a matrix behave similarly

fractal properties of invariant subsets for piecewise monotonic maps on the interval let t be a piecewise monotonic transformation on 0 1 and let a be a t invariant subset which has positive topological entropy and satisfies the darboux property a general existence theorem for conformal measures on a is proved this is then used to show equality of the dynamical dimension of a and the minimal zero of a certain pressure function

topological markov chains c algebras and k2 soit a x a x a la chaine de markov topologique provenant d une matrice 0 1 pouvant etre infinie et soit aut a le groupe des equivalences uniformes de a on etudie la relation entre aut a le foncteur algebrique de k theorie k 2 et des automorphismes de c algebre du type cuntz kreiger associees a a et de son groupe de dimension correspondant k 0 x a

conditionally invariant measures and exponential decay let a measure u be given in a what can be said about the distribution of n and does the measure p tp e p t a converge to some limit measure pm e and if it does is pu independent of the initial measure u and what are the properties of p pianigiani and yorke 5 studied extensively such problems for mappings t which are expansive in a sense stated below in particular they show the existence and the uniqueness of a smooth uu and they prove that the distribution of n is nearly exponential in the first part of this paper we study the above questions for maps which are not expansive in the second part of the paper we study systems depending upon a parameter such systems arise naturally from physical and biological problems for example the lorenz model for a fluid flow is a system of differential equations depending on a parameter r the rayleigh number associated with this model is a family of maps t 0 l ir see 6 interesting properties of the original system can be understood in terms of these maps let t a ir a c ir e e 0 eo e 0 be a family of maps and for each e e 0 so let n x be the kickout time function define d r 0 l by

resonances of dynamical systems and fredholm riesz operators on rigged hilbert spaces abstract resonances of dynamical systems are defined as the singularities of the analytically continued resolvent of the restriction of the frobenius perron operator to suitable test function spaces a sufficient condition for resonances to arise from a meromorphic continuation to the entire plane is that the frobenius perron operator is a fredholm riesz operator on a rigged hilbert space after a discussion of spectral theory in locally convex topological vector spaces we illustrate the approach for a simple chaotic system namely the renyi map

stochastically stable globally coupled maps with bistable thermodynamic limit we study systems of globally coupled interval maps where the identical individual maps have two expanding fractional linear onto branches and where the coupling is introduced via a parameter common to all individual maps that depends in an analytic way on the mean field of the system we show 1 for the range of coupling parameters we consider finite size coupled systems always have a unique invariant probability density which is strictly positive and analytic and all finite size systems exhibit exponential decay of correlations 2 for the same range of parameters the self consistent perron frobenius operator which captures essential aspects of the corresponding infinite size system arising as the limit of the above when the system size tends to infinity undergoes a supercritical pitchfork bifurcation from a unique stable equilibrium to the coexistence of two stable and one unstable equilibrium

probabilistic models of multidimensional piecewise expanding mappings strongly chaotic systems e g piecewise expanding mappings exhibit diffusion like behavior in the sense of central limit theorems to find more precise statements about the similarity to probabilistic diffusion we study how the evolution of probability densities underd dimensional piecewise expanding mappings can be modeled by markov processes with smooth transition probabilities such as diffusion processes our results can be viewed as a special type of local limit theorem

ergodic properties of markov maps inr d for domainsi r d bounded or not the notion of a markov map fromi into itself is developed it is shown that under a condition of renyi type and the assumption that the map is markov any probability density tends inl1 norm to a unique invariant measure under the action of the perron frobenius operatorp the smoothness and ergodic properties of that invariant measure are studied the paper generalizes results of lasota and yorke from dimension one to higher dimension

symbolic dynamics for surface diffeomorphisms with positive entropy part 1 chains as pseudo orbits 8 2 pesin charts 8 2 1 non uniform hyperbolicity 8 2 2 lyapunov change of coordinates 8 2 3 pesin charts 10 2 4 distortion compensating bounds 11 2 5 nuh f 11 3 overlapping charts 11 3 1 the overlap condition 12 3 2 the form of f in overlapping charts 14 3 3 coarse graining 16 4 e chains and an infinite to one markov extension of f 18 4 1 double charts and e chains 18 4 2 admissible manifolds and the graph transform 20 4 3 a markov extension 23 4 4 the relevant part of the extension 27

on the rate of convergence to equilibrium in one dimensional systems we determine the essential spectral radius of the perron frobenius operator for piecewise expanding transformations considered as an operator on the space of functions of bounded variation and relate the speed of convergence to equilibrium in such one dimensional systems to the greatest eigenvalues of generalized perron frobenius operators of the transformations operators which yield singular invariant measures

improved lower bounds on the ground state entropy of the antiferromagnetic potts model we present generalized methods for calculating lower bounds on the ground state entropy per site s 0 or equivalently the ground state degeneracy per site w e s 0 k b of the antiferromagnetic potts model we use these methods to derive improved lower bounds on w for several lattices

exponential inequalities for dynamical measures of expanding maps of the interval we prove an exponential inequality for the absolutely continuous invariant measure of a piecewise expanding map of the interval as an immediate corollary we obtain a concentration inequality we apply these results to the estimation of the rate of convergence of the empirical measure in various metrics and also to the efficiency of the shadowing by sets of positive measure

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

spectral theory zeta functions and the distribution of periodic points for collet eckmann maps we study unimodal interval mapst with negative schwarzian derivative satisfying the collet eckmann condition dtn tc k cn for some constantsk 0 and c 1 c is the critical point oft we prove exponential mixing properties of the unique invariant probability density oft describe the long term behaviour of typical in the sense of lebesgue measure trajectories by central limit and large deviations theorems for partial sum processes of the form s n sigma i 0 n 1 f t i x and study the distribution of typical periodic orbits also in the sense of a central limit theorem and a large deviations theorem

a nonequilibrium entropy for dynamical systems it is proposed to define entropy for nonequilibrium ensembles using a method of coarse graining which partitions phase space into sets which typically have zero measure these are chosen by considering the totality of future possibilities for observation on the system it is shown that this entropy is necessarily a nondecreasing function of the timet there is no contradiction with the reversibility of the laws of motion because this method of coarse graining is asymmetric under time reversal under suitable conditions which are stated explicitly this entropy approaches the equilibrium entropy ast and the fine grained entropy ast in particular the conditions can always be satisfied if the system is ak system as in the sinai billiard models some theorems are given which give information about whether it is possible to generate the partition used here for coarse graining from time translates of a finite partition and at the same time elucidate the connection between our concept of entropy and the entropy invariant of kolmogorov and sinai

a local limit theorem for closed geodesics and homology in this paper we study the distribution of closed geodesics on a compact negatively curved manifold we concentrate on geodesics lying in a prescribed homology class and under certain conditions obtain a local limit theorem to describe the asymptotic behaviour of the associated counting function as the homology class varies

compact locally maximal hyperbolic sets for smooth maps fine statistical properties compact locally maximal hyperbolic sets are studied via geometrically defined functional spaces that take advantage of the smoothness of the map in a neighborhood of the hyperbolic set this provides a self contained theory that not only reproduces all the known classical results but also gives new insights on the statistical properties of these systems

on the relation between lyapunov exponents and exponential decay of correlations chaotic dynamics with sensitive dependence on initial conditions may result in exponential decay of correlation functions we show that for one dimensional interval maps the corresponding quantities that is lyapunov exponents and exponential decay rates are related more specifically for piecewise linear expanding markov maps observed via piecewise analytic functions we show that the decay rate is bounded above by twice the lyapunov exponent that is we establish lower bounds for the subleading eigenvalue of the corresponding perron frobenius operator in addition we comment on similar relations for general piecewise smooth expanding maps

annealed and quenched limit theorems for random expanding dynamical systems in this paper we investigate annealed and quenched limit theorems for random expanding dynamical systems making use of functional analytic techniques and more probabilistic arguments with martingales we prove annealed versions of a central limit theorem a large deviation principle a local limit theorem and an almost sure invariance principle we also discuss the quenched central limit theorem dynamical borel cantelli lemmas erdos renyi laws and concentration inequalities

ergodic properties of invariant measures for piecewise monotonic transformations during the last decade a lot of research has been done on onedimensional dynamics different kinds of invariant measures for certain classes of piecewise monotonic transformations have been considered in most of these cases the perron frobenius operator plays an important role in this paper we try to unify these different examples which are discussed in detail below first we give a discription of the results proved in this paper

large deviations for poisson systems of independent random walks we prove large deviation theorems for occupation time functionals of independent random walks started from a poisson field on zd in dimensions 1 and 2 the large deviation tails are larger than exponential exact asymptotics are derived

central limit theorem and stable laws for intermittent maps in the setting of abstract markov maps we prove results concerning the convergence of renormalized birkhoff sums to normal laws or stable laws they apply to one dimensional maps with a neutral fixed point at 0 of the form x x1 for 0 1 in particular for 1 2 we show that the birkhoff sums of a holder observable f converge to a normal law or a stable law depending on whether f 0 0 or f 0 0 the proof uses spectral techniques introduced by sarig and wiener s lemma in non commutative banach algebras

dynamical entropy for systems with stochastic perturbation dynamics of deterministic systems perturbed by random additive noise is characterized quantitatively since for such systems the kolmogorov sinai ks entropy diverges if the diameter of the partition tends to zero we analyze the difference between the total entropy of a noisy system and the entropy of the noise itself we show that this quantity is finite and non negative and we call it the dynamical entropy of the noisy system in the weak noise limit this quantity is conjectured to tend to the ks entropy of the deterministic system in particular we consider one dimensional systems with noise described by a finite dimensional kernel for which the frobenius perron operator can be represented by a finite matrix

piecewise invertible dynamical systems the aim of the paper is the investigation of piecewise monotonic maps t of an interval x the main tool is an isomorphism of x t with a topological markov chain with countable state space which is described by a 0 1 transition matrix m the behavior of the orbits of points in x under t is very similar to the behavior of the paths of the markov chain every irreducible submatrix of m gives rise to a t invariant subset l of x such that l is the set x of all limit points of the orbit of an x x the topological entropy of l is the logarithm of the spectral radius of the irreducible submatrix which is a l 1 operator besides these sets l there are two t invariant sets y and p such that for all x x the set x is either contained in one of the sets l or in y or in p the set p is a union of periodic orbits and y is contained in a finite union of sets y with y x and has topological entropy zero this isomorphism of x t with a topological markov chain is also an important tool for the investigation of t invariant measures on x results in this direction which are published elsewhere are described at the end of the paper furthermore a part of the proofs in the paper is purely topological without using the order relation of the interval x so that some results hold for more general dynamical systems x t

a spectral approach for quenched limit theorems for random hyperbolic dynamical systems we extend the recent spectral approach for quenched limit theorems developed for piecewise expanding dynamics under general random driving to quenched random piecewise hyperbolic dynamics for general ergodic sequences of maps in a neighborhood of a hyperbolic map we prove a quenched large deviations principle ldp central limit theorem clt and local central limit theorem lclt

zeta functions and transfer operators for piecewise monotone transformations given a piecewise monotone transformationt of the interval and a piecewise continuous complex weight functiong of bounded variation we prove that the ruelle zeta function z of t g extends meromorphically to z 1 where lim g tn 1 g tg 1 n and thatz is a pole of if and only ifz 1 is an eigenvalue of the corresponding transfer operator l we do not assume that l leaves a reference measure invariant

estimating a function from ergodic samples with additive noise we study the problem of estimating an unknown function from ergodic samples corrupted by additive noise it is shown that one can consistently recover an unknown measurable function in this setting if the one dimensional 1 d distribution of the samples is comparable to a known reference distribution and the noise is independent of the samples and has known mixing rates the estimates are applied to deterministic sampling schemes in which successive samples are obtained by repeatedly applying a fixed map to a given initial vector and it is then shown how the estimates can be used to reconstruct an ergodic transformation from one of its trajectories

strange attractors and asymptotic measures of discrete time dissipative systems we investigate asymptotic properties of certain discrete time dynamical systems in two and three dimensions with solenoidal attractor it is proved that the asymptotic measures relevant for the generalized version of the ergodic theorem all derive from one haar measure

a spectral approach for quenched limit theorems for random expanding dynamical systems we prove quenched versions of i a large deviations principle ldp ii a central limit theorem clt and iii a local central limit theorem lclt for non autonomous dynamical systems a key advance is the extension of the spectral method commonly used in limit laws for deterministic maps to the general random setting we achieve this via multiplicative ergodic theory and the development of a general framework to control the regularity of lyapunov exponents of twisted transfer operator cocycles with respect to a twist parameter while some versions of the ldp and clt have previously been proved with other techniques the local central limit theorem is to our knowledge a completely new result and one that demonstrates the strength of our method applications include non autonomous piecewise expanding maps defined by random compositions of the form t n 1 t t an important aspect of our results is that we only assume ergodicity and invertibility of the random driving in particular no expansivity or mixing properties are required

scaling properties of ℤk 1 actions on the circle we derive universal scaling properties for k 1 actions on the circle whose generators have rotation numbers algebraic of degreek as fork 2 these properties can be explained for arbitraryk in terms of a renormalization group transformation it has at least one trivial fixed point corresponding to an action whose generators are pure rotations the spectrum of the linearized transformation in this fixed point is analyzed completely the fixed point is hyperbolic with a k 1 dimensional unstable manifold in the casek 2 the known results are therefore recovered

on the susceptibility function of piecewise expanding interval maps we study the susceptibility function psi z sum n 0 infty int z n x y rho 0 y frac partial partial y varphi f n y dy associated to the perturbation f t f tx circ f of a piecewise expanding interval map f and to an observable 1 is the formal derivative at t 0 of the average mathcal r t int varphi rho t dx of with respect to the srb measure of ft our analysis is based on a spectral description of transfer operators it gives in particular sufficient conditions on f x and which guarantee that z is holomorphic in a disc of larger than one or which ensure that a number may be associated to the possibly divergent series 1 we present examples of f x and so that mathcal r t is not lipschitz at 0 and we propose a new version of ruelle s conjecture

discretization of the frobenius perron operator using a sparse haar tensor basis the sparse ulam method the global macroscopic behavior of a dynamical system is encoded in the eigenfunctions of the associated frobenius perron operator for systems with low dimensional long term dynamics efficient techniques exist for a numerical approximation of the most important eigenfunctions cf m dellnitz and o junge siam j numer anal 36 1999 pp 491 515 they are based on a projection of the operator onto a space of piecewise constant functions supported on a neighborhood of the attractor ulam s method in this paper we develop a numerical technique which makes ulam s approach applicable to systems with higher dimensional long term dynamics it is based on ideas for the treatment of higher dimensional partial differential equations using sparse grids c zenger sparse grids in parallel algorithms for partial differential equations kiel 1990 vieweg braunschweig 1991 pp 241 251 h j bungartz and m griebel acta numer 13 2004 pp 147 269 here we use a sparse haar tensor basis as the underlying approximation space we develop the technique establish statements about its complexity and convergence and present two numerical examples

on the prevalence of horseshoes in this paper the symbolic dynamics of several differentiable systems are investigated it is shown that many well known dynamical systems including axiom a systems piecewise monotonic maps of the interval the lorenz attractor and abraham smale examples have inside them subsystems conjugate to subshifts of finite type these subsystems have hyperbolic structures and hence are stable they can also be chosen to have entropy arbitrarily close to that of the ambient system let f m m be a diffeomorphism of a manifold into itself and let q f be its nonwandering set while it is usually difficult to describe completely the orbit structure of f one could look for large invariant subsets of q f on which the dynamics of f are simple to characterize this idea is due to rufus bowen who proposed that one asks the following given e 0 does there exist a hyperbolic f invariant set a c 2 f with the property that l fja is conjugate to a subshift of finite type and 2 h f iae h f e where h f is the topological entropy of f we say that f is in the sense of entropy a limit of hyperbolic subshifts of finite type if a above exists for every 0 the main result of this paper is that many dynamical systems we know are limits of this kind they include axiom a diffeomorphisms and flows piecewise monotonic maps of the interval the poincare map of the lorenz attractor 15 and certain abraham smale examples 1 perhaps this new class of systems accounts for some of the other non axiom a examples as well e g 11 17 19 23 24 28 but that remains to be decided recall the definition of a subshift of finite type let 1 n be given the discrete topology and l i 1 n the product topology let a l e be defined by ax i x i where x i denotes the ith coordinate of x let a au be annxn matrix of o s and l s let ea xel axx 1 for alliez then ai a is called a subshift of finite type if 2 l 1 n and y x e y a i 1 for all i 0 then aiya is called a one sided subshift of finite type from now on subshifts of finite type will be abbreviated as ssft received by the editors january 26 1979 ams mos subject classifications 1970 primary 58f15 secondary 28a65

l rays of permutation matrices and doubly stochastic matrices abstract let m n be the space of real n n matrices we investigate a linear transformation m n r n called an l ray motivated by x ray which is defined in terms of sums of the entries in the blocks of a certain l shaped partition of the positions of a matrix a m n we find descriptions of the image of the classes of permutation matrices and doubly stochastic matrices under this map and show connections to majorization theory

specification on the interval we study the consequences of discontinuities on the specification property for interval maps after giving a necessary and sufficient condition for a piecewise monotonic piecewise continuous map to have this property we show that for a large and natural class of families of such maps including the 3 transformations the set of parameters for which the specification property holds though dense has zero lebesgue measure thus regarding the specification property the general case is at the opposite of the continuous case solved by a m blokh russian math surveys 38 1983 133 134 for which we give a proof

on the approximation of invariant measures given a discrete dynamical system defined by the map x x the density of the absolutely continuous a c invariant measure if it exists is the fixed point of the frobenius perron operator defined on l1 x ulam proposed a numerical method for approximating such densities based on the computation of a fixed point of a matrix approximation of the operator t y li proved the convergence of the scheme for expanding maps of the interval g keller and m blank extended this result to piecewise expanding maps of the cube in n we show convergence of a variation of ulam s scheme for maps of the cube for which the frobenius perron operator is quasicompact we also give sufficient conditions on for the existence of a unique fixed point of the matrix approximation and if the fixed point of the operator is a function of bounded variation we estimate the convergence rate

diffusion effusion and chaotic scattering an exactly solvable liouvillian dynamics we study diffusion and chaotic scattering in a chain of baker maps coupled together which forms an area preserving mapping of an infinitely extended strip onto itself this exactly solvable mapping sustains chaotic behaviors and diffusion processes the relationship between the diffusion coefficient the lyapunov exponent and the entropy per unit time is derived the long lived classical resonances of the liouville evolution operator are proved to converge toward the eigenvalues of the phenomenological diffusion equation in this sense there is a quasi isomorphism between the resonance spectrum of the liouville evolution and the eigenvalue spectrum of the phenomenological diffusion equation furthermore we show that a fractal repeller is associated to each non equilibrium state in the isolated and finite multibaker chain the nonequilibrium states are all unstable with respect to the equilibrium validating a weak form of the second principle of thermodynamics for the present dynamical system consequences of nonequilibrium fractals on classical measurements are discussed we then describe the open multibaker chain as a scattering system fractal properties of chaotic scattering are here shown to be related to diffusion in the chain

computing the topological entropy for piecewise monotonic maps on the interval a new method for computing the topological entropy of a piecewise monotonic transformation on the interval is presented it uses a transition matrix associated with the transformation for this matrix we give a spectral theorem this can be used for an estimation of the accuracy of the algorithm

rigorous statistical procedures for data from dynamical systems various questions about the invariant measures of a dynamical system can be answered by computations of regular functionals or by ranking methods based on a set of observations this includes symmetry tests and the determination of dimension coefficients the paper contains the theoretical results and several simulations explain the methods

approach to equilibrium for locally expanding maps in ℝ k by using a well known technique from classical statistical mechanics of one dimensional lattice spin systems we prove existence of an absolutely continuous invariant asymptotic measure for certain locally expanding mapst of the unit cube in k we generalize herewith in a certain sense the results of lasota and yorke on piecewise expanding maps of the unit interval to higher dimensions we show a kuzmin type theorem for these systems from which exponential approach to equilibrium and strong mixing properties follow

on the relaxation time of gauss continued fraction map ii the banach space approach transfer operator method the spectrum of the transfer operator for the maptx 1 x 1 x when restricted to a certain banach space of holomorphic functions is shown to coincide with the spectrum of the adjointu of koopman s isometric operatoruf x f t x when the former is restricted to the hilbert space introduced in part i of this work ifn denotes the operator p1 withp1 the projector onto the eigenfunction to the dominant eigenvalue 1 1 of then n is au0 positive operator with respect to some cone and therefore has a dominant positive simple eigenvalue 2 a minimax principle holds giving rigorous upper and lower bounds both for 2 and the relaxation time of the mapt

continued fractions and the d dimensional gauss transformation in this paper we study a multidimensional continued fraction algorithm which is related to the modified jacobi perron algorithm considered by podsypanin and schweiger we demonstrate that this algorithm has many important properties which are natural generalisations of properties of one dimensional continued fractions for this reason we call the transformation associated to the algorithm the d dimensional gauss transformation we construct a coordinate system for the natural extension which reveals its symmetries and allows one to give an explicit formula for the density of its invariant measure we also discuss the ergodic properties of this invariant measure

what you need to know to knead 1 introduction 2 the main kneading identity 3 milnor and thurston s definitions 4 an identity for the kneading determinant 5 the inverse of the kneading matrix 6 meromorphic functions in the unit disk 7 piecewise linear models 8 counting fixed points 9 total variation 10 an invariant measure with maximal entropy 11 appendix kneading theory for mappings with discontinuities

correlations of length spectra for negatively curved manifolds in this paper we obtain asymptotic estimates for pairs of closed geodesics on negatively curved manifolds the differences of whose lengths lie in a prescribed family of shrinking intervals where the geodesics are ordered with respect to a discrete length in certain cases this discrete length can be taken to be the word length with respect to a set of generators for the fundamental group

almost isomorphism for countable state markov shifts countable state markov shifts are a natural generalization of the well known subshifts of finite type they are the subject of current research both for their own sake and as models for smooth dynamical systems in this paper we investigate their almost isomorphism and entropy conjugacy and obtain a complete classification for the especially important class of strongly positive recurrent markov shifts this gives a complete classification up to entropyconjugacy of the natural extensions of smooth entropy expanding maps e g c smooth interval maps with non zero topological entropy

mixing properties in expanding lorenz maps abstract let t f 0 1 0 1 be an expanding lorenz map this means t f x f x mod 1 where f 0 1 0 2 is a strictly increasing map satisfying inf f 1 then t f has two pieces of monotonicity in this paper sufficient conditions when t f is topologically mixing are provided for the special case f x x with 2 3 a full characterization of parameters leading to mixing is given furthermore relations between renormalizability and t f being locally eventually onto are considered and some gaps in classical results on the dynamics of lorenz maps are corrected

natural equilibrium states for multimodal maps this paper is devoted to the study of the thermodynamic formalism for a class of real multimodal maps this class contains but it is larger than collet eckmann for a map in this class we prove existence and uniqueness of equilibrium states for the geometric potentials t log df for the largest possible interval of parameters t we also study the regularity and convexity properties of the pressure function completely characterising the first order phase transitions results concerning the existence of absolutely continuous invariant measures with respect to the lebesgue measure are also obtained

meromorphic extensions of generalised zeta functions in this paper we give a full description of the spectrum of the ruelle perron frobenius operator acting on the banach space of holder continuous functions on a subshift of finite type theorem 1 these results are then used to extend the meromorphic domain of generalised zeta functions theorem 2 the most important application of these results is to the domain of the smale zeta function for axiom a flows theorem 3 in the course of this paper we settle questions raised by ruelle and sunada

local limit theorems for free groups in this paper we obtain a local limit theorem for elements of a free group g under the abelianization map cdot g to g g g this is obtained via an analysis involving subshifts of finite type where we obtain a result of independent interest the case of fundamental groups of compact surfaces of genus geq 2 is also discussed

mixing rates and exterior forms in chaotic systems the authors propose a numerical method for extracting the spectrum of decay rates of time correlations in chaotic dynamical systems the sum of the first p decay rates is related to the asymptotic behaviour of the time correlation of suitable exterior forms of order p the method is applied to maps of the interval and to the henon map