higher order correlations of tchebyscheff maps we determine the higher order correlation functions of iterates of tchebyscheff polynomials of arbitrary order n the non vanishing correlations can be described by suitable sets of graphs these graphs allow for a perturbative evaluation of invariant densities for langevin like systems that are driven by deterministic chaotic forces generated by tchebyscheff maps rather than gaussian white noise

central limit theorems and invariance principles for time one maps of hyperbolic flows we give a general method for deducing statistical limit laws in situations where rapid decay of correlations has been established as an application of this method we obtain new results for time one maps of hyperbolic flows

nice inducing schemes and the thermodynamics of rational maps we study the thermodynamic formalism of a complex rational map f of degree at least two viewed as a dynamical system acting on the riemann sphere more precisely for a real parameter t we study the existence of equilibrium states of f for the potential t rm ln left f prime right and the analytic dependence on t of the corresponding pressure function we give a fairly complete description of the thermodynamic formalism for a large class of rational maps including well known classes of non uniformly hyperbolic rational maps such as topological collet eckmann maps and much beyond in fact our results apply to all non renormalizable polynomials without indifferent periodic points to infinitely renormalizable quadratic polynomials with a priori bounds and all quadratic polynomials with real coefficients as an application for these maps we describe the dimension spectrum for lyapunov exponents and for pointwise dimensions of the measure of maximal entropy and obtain some level 1 large deviations results for polynomials as above we conclude that the integral means spectrum of the basin of attraction of infinity is real analytic at each parameter in mathbb r with at most two exceptions

equilibrium states of weakly hyperbolic one dimensional maps for holder potentials there is a wealth of results in the literature on the thermodynamic formalism for potentials that are in some sense hyperbolic we show that for a sufficiently regular one dimensional map satisfying a weak hyperbolicity assumption every holder continuous potential is hyperbolic a sample consequence is the absence of phase transitions the pressure function is real analytic on the space of holder continuous functions another consequence is that every holder continuous potential has a unique equilibrium state and that this measure has exponential decay of correlations

central limit theorems for additive functionals of markov chains central limit theorems and invariance principles are obtained for additive functionals of a stationary ergodic markov chain say s n g x 1 g x n where e g x 1 0 and e g x 1 2 the conditions imposed restrict the moments of g and the growth of the conditional means e s n x 1 no other restrictions on the dependence structure of the chain are required when specialized to shift processes the conditions are implied by simple integral tests involving g

statistical properties of dynamical systems with some hyperbolicity this paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts though not necessarily all of their phase spaces the main results are for discrete time systems to put this work into context recall that for axiom a attractors the picture has been fairly complete since the 1970 s see s1 b r2 since then much progress has been made on two fronts there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero lyapunov exponents o p1 ka ly and there are detailed analyses of specific kinds of dynamical systems including for example billiards 1 dimensional and henon type maps s2 bsc hk j bc2 by1 statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero lyapunov exponents the goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than axiom a this class will not be defined explicitly but it includes some of the much studied examples by looking at regular returns to sets with good hyperbolic properties one could give systems in this class a simple dynamical representation conditions for the existence of natural invariant measures exponential mixing and central limit theorems are given in terms of the return times these conditions can be checked in concrete situations giving a unified way of proving a number of results some new and some old among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of henon type maps

rapid mixing for the lorenz attractor and statistical limit laws for their time 1 maps we prove that every geometric lorenz attractor satisfying a strong dissipativity condition has superpolynomial decay of correlations with respect to the unique sinai ruelle bowen measure moreover we prove the central limit theorem and almost sure invariance principle for the time 1 map of the flow of such attractors in particular our results apply to the classical lorenz attractor

local convexity and local boundedness of induced i l topological vector spaces in this paper we prove that an l topological vector space is locally convex locally bounded iff its induced i l topological vector space is locally convex locally bounded where i l is the l fuzzy unit interval also we give a necessary and sufficient condition for the induced i l topological vector space to be i l fuzzy normable

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

the central limit theorem for markov chains with normal transition operators started at a point the central limit theorem and the invariance principle proved by kipnis and varadhan for reversible stationary ergodic markov chains with respect to the stationary law are established with respect to the law of the chain started at a fixed point almost surely under a slight reinforcing of their spectral assumption the result is valid also for stationary ergodic chains whose transition operator is normal

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

fragment length distributions and collision probabilities for aflp markers aflp is a dna fingerprinting technique frequently used in plant and animal sciences a drawback of the technique is the occurrence of multiple dna fragments of the same length in a single aflp lane which we name a collision in this article we quantify the problem the well known birthday problem plays a role calculation of collision probabilities requires a fragment length distribution fld we discuss three ways to estimate the fld based on theoretical considerations on in silico determination using dna sequence data from arabidopsis thaliana or on direct estimation from aflp data in the latter case we use a generalized linear model with monotone smoothing of the fragment length probabilities collision probabilities are calculated from two perspectives assuming known fragment counts and assuming known band counts we compare results for a number of fld s ranging from uniform to highly skewed the conclusion is that collisions occur often with higher probabilities for higher numbers of bands for more skewed distributions and to a lesser extent for smaller scoring ranges for a typical plant genome an aflp with 19 bands is likely to contain the first collision practical implications of collisions are discussed aflp examples from lettuce and chicory are used for illustration

an invariance principle for maps with polynomial decay of correlations we give a general method of deriving statistical limit theorems such as the central limit theorem and its functional version in the setting of ergodic measure preserving transformations this method is applicable in situations where the iterates of discrete time maps display a polynomial decay of correlations

subexponential decay of correlations we describe a method for proving subexponential lower bounds for correlations functions and apply it to study decay of correlations for maps with countable markov partitions one result is that ls young s upper estimates y2 are optimal in many situations our method is based on a general result concerning the asymptotics of renewal sequences of bounded operators acting on banach spaces which we apply to the iterates of the transfer operator

renewal sequences and intermittency in this paper we examine the generating function z of a renewal sequence arising from the distribution of return times in the turbulent region for a class of piecewise affine interval maps introduced by gaspard and wang and studied by several authors we prove that it admits a meromorphic continuation to the entire complex z plane with a branch cut along the ray 1 moreover we compute the asymptotic behavior of the coefficients of its taylor expansion at z 0 from this we obtain the exact polynomial asympotics for the rate of mixing when the invariant measure is finite and of the scaling rate when it is infinite

limit theorems for partially hyperbolic systems we consider a large class of partially hyperbolic systems containing among others affine maps frame flows on negatively curved manifolds and mostly contracting diffeomorphisms if the rate of mixing is sufficiently high the system satisfies many classical limit theorems of probability theory

limit theorems and markov approximations for chaotic dynamical systems we prove the central limit theorem and weak invariance principle for abstract dynamical systems based on bounds on their mixing coefficients we also develop techniques of markov approximations for dynamical systems we apply our results to expanding interval maps axiom a diffeomorphisms chaotic billiards and hyperbolic attractors

rate of convergence in the weak invariance principle for deterministic systems we obtain the first results on convergence rates in the prokhorov metric for the weak invariance principle functional central limit theorem for deterministic dynamical systems our results hold for uniformly expanding hyperbolic axiom a systems as well as nonuniformly expanding hyperbolic systems such as dispersing billiards henon like attractors viana maps and intermittent maps as an application we obtain convergence rates for deterministic homogenization in multiscale systems

on ruelle perron frobenius operators i ruelle theorem we study ruelle perron frobenius operators for locally expanding and mixing dynamical systems on general compact metric spaces associated with potentials satisfying the dini condition in this paper we give a proof of the ruelle theorem on gibbs measures it is the first part of our research on the subject the rate of convergence of powers of the operator will be presented in a forthcoming paper

on ruelle perron frobenius operators ii convergence speeds we study ruelle operators on expanding and mixing dynamical systems with potential function satisfying the dini condition we give an estimate for the convergence speed of the iterates of a ruelle operator our proof avoids markov partitions this is the second part of our research on ruelle operators

a representation for multinomial cumulative distribution functions a re expression of the usual representation of the multinomial distribution as the conditional distribution of independent poisson random variables given fixed sum provides a convenient new way to compute multinomial cumulative distribution functions

the central limit theorem for markov chains started at a point the aim of this paper is to prove a central limit theorem and an invariance principle for an additive functional of an ergodic markov chain on a general state space with respect to the law of the chain started at a point no irreducibility assumption nor mixing conditions are imposed the only assumption bears on the growth of the l 2 norms of the ergodic sums for the function generating the additive functional which must be with the result holds almost surely with respect to the invariant probability of the chain

decay of correlations dedicated to micheline ishay i would like to thank p boyland l chierchia v donnay g de martino c gole j l lebowitz m lyubich m rychlik i g schwarz s vaienti and especially g gallavotti for helpful and enlightening discussions particularly warm thanks go to n chernov for carefully reading and finding a mistake in an early version the present paper benefits from several improvements due to his sharp criticism in addition i am indebted to p collet and most of all m wojtkowski for introducing me to the magical world of cones finally i thank j milnor director of the institute for mathematical sciences at stony brook university where i was visiting during part of this work and the italian c n r gnfm for providing travel funds

invariance principles for interval maps with an indifferent fixed point in this note we establish an almost sure invariance principle for a large class of interval maps with indifferent fixed points including the manneville pomeau map this implies a number of well known corollaries including the weak invariance principle and the law of the iterated logarithm