maps of intervals with indifferent fixed points thermodynamic formalism and phase transitions we develop the thermodynamic formalism for a large class of maps of the interval with indifferent fixed points for such systems the formalism yields onedimensional systems with many body infinite range interactions for which the thermodynamics is well defined but gibbs states are not piecewise linear systems of this kind yield the soluble in a sense fisher models we prove that such systems exhibit phase transitions the order of which depends on the behavior at the indifferent fixed points we obtain the critical exponent describing the singularity of the pressure and analyze the decay of correlations of the equilibrium states at all temperatures our technique relies on establishing and exploiting a relation between the transfer operators of the original map and its suitable expanding induced version the technique allows one also to obtain a version of the bowen ruelle formula for the hausdorff dimension of repellers for maps with indifferent fixed points and to generalize fisher results to some nonsoluble models

analysis of parameter selections for fuzzy c means the weighting exponent m is called the fuzzifier that can influence the performance of fuzzy c means fcm it is generally suggested that email protected 1 5 2 5 on the basis of a robust analysis of fcm a new guideline for selecting the parameter m is proposed we will show that a large m value will make fcm more robust to noise and outliers however considerably large m values that are greater than the theoretical upper bound will make the sample mean a unique optimizer a simple and efficient method to avoid this unexpected case in fuzzy clustering is to assign a cluster core to each cluster we will also discuss some clustering algorithms that extend fcm to contain the cluster cores in fuzzy clusters for a large theoretical upper bound case we suggest the implementation of the fcm with a suitable large m value otherwise we suggest implementing the clustering methods with cluster cores when the data set contains noise and outliers the fuzzifier m 4 is recommended for both fcm and cluster core based methods in a large theoretical upper bound case

statistical properties for flows with unbounded roof function including the lorenz attractor for geometric lorenz attractors including the classical lorenz attractor we obtain a greatly simplified proof of the central limit theorem which applies also to the more general class of codimension two singular hyperbolic attractors we also obtain the functional central limit theorem and moment estimates as well as iterated versions of these results a consequence is deterministic homogenisation convergence to a stochastic differential equation for fast slow dynamical systems whenever the fast dynamics is singularly hyperbolic of codimension two

a vector valued almost sure invariance principle for hyperbolic dynamical systems we prove an almost sure invariance principle approximation by d dimensional brownian motion for vector valued holder observables of large classes of nonuniformly hyperbolic dynamical systems these systems include axiom a dieomorphisms and flows as well as systems modelled by young towers with moderate tail decay rates in particular the position variable of the planar periodic lorentz gas with finite horizon approximates a 2 dimensional brownian motion

a novel subgridding scheme based on a combination of the finite element and finite difference time domain methods a simple and versatile local mesh refinement scheme based on the hybridization of the finite element fe and the finite difference time domain fdtd algorithms is presented the scheme achieves considerable flexibility in subgridding by using a transition region between the coarse and fine fdtd grids meshed according to an unstructured grid and solved by means of the fe method in the td an interpolation scheme in the time domain which allows the use of different time steps in the coarse and fine mesh regions is included in the paper

yet another induction scheme for non uniformly expanding transformations abstract we introduce a new induction scheme for non uniformly expanding maps f of compact riemannian manifolds relying upon ideas of 33 and 10 we use this induction approach to prove that the existence of a gibbs markov young structure is a necessary condition for f to preserve an absolutely continuous probability with all its lyapunov exponents positive

from rates of mixing to recurrence times via large deviations abstract a classic approach in dynamical systems is to use particular geometric structures to deduce statistical properties for example the existence of invariant measures with stochastic like behaviour such as large deviations or decay of correlations such geometric structures are generally highly non trivial and thus a natural question is the extent to which this approach can be applied in this paper we show that in many cases stochastic like behaviour itself implies that the system has certain non trivial geometric properties which are therefore necessary as well as sufficient conditions for the occurrence of the statistical properties under consideration as a by product of our techniques we also obtain some new results on large deviations for certain classes of systems which include viana maps and multidimensional piecewise expanding maps

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

fdtd local grid with material traverse often a finite difference time domain fdtd calculation requires a relatively higher mesh resolution in only small subvolumes of the total mesh space by locally applying finer grids local grids to these volumes the necessary resolution can be obtained computation time and memory requirements may be far less than for an fdtd space with the smaller mesh resolution throughout in many situations it is important that these local grids function when materials traverse the main grid local grid mg lg boundary surfaces since the volumes that require local grids may not be isolated in a homogeneous medium a local grid method which allows dielectric and or conducting materials to traverse the boundaries is developed three different fdtd problems that utilize the local grid method are used as validation tests results are compared with uniform mesh fdtd solutions

geometric and measure theoretical structures of maps with mostly contracting center we show that every diffeomorphism with mostly contracting center direction exhibits a geometric combinatorial structure which we call skeleton that determines the number basins and supports of the physical measures furthermore the skeleton allows us to describe how the physical measures bifurcate as the diffeomorphism changes in particular we use this to construct examples with any given number of physical measures with basins densely intermingled and to analyse how these measures collapse into each other through explosions of their basins as the dynamics varies this theory also allows us to prove that in the absence of collapses the basins are continuous functions of the diffeomorphism

equidistribution for nonuniformly expanding dynamical systems and application to the almost sure invariance principle let t m mt m m be a nonuniformly expanding dynamical system such as logistic or intermittent map let v m rdv m rd be an observable and vn n 1k 0v tkvn k 0n 1v tk denote the birkhoff sums given a probability measure on m we consider vn as a discrete time random process on the probability space m m in smooth ergodic theory there are various natural choices of such as the lebesgue measure or the absolutely continuous t invariant measure they give rise to different random processes we investigate relation between such processes we show that in a large class of measures it is possible to couple redefine on a new probability space every two processes so that they are almost surely close to each other with explicit estimates of closeness the purpose of this work is to close a gap in the proof of the almost sure invariance principle for nonuniformly hyperbolic transformations by melbourne and nicol

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

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

large deviations for nonuniformly hyperbolic systems we obtain large deviation estimates for a large class of nonuniformly hyperbolic systems namely those modelled by young towers with summable decay of correlations in the case of exponential decay of correlations we obtain exponential large deviation estimates given by a rate function in the case of polynomial decay of correlations we obtain polynomial large deviation estimates and exhibit examples where these estimates are essentially optimal in contrast with many treatments of large deviations our methods do not rely on thermodynamic formalism hence for holder observables we are able to obtain exponential estimates in situations where the space of equilibrium measures is not known to be a singleton as well as polynomial estimates in situations where there is not a unique equilibrium measure

invariance principles for iterated maps that contract on average we consider iterated function schemes that contract on average using a transfer operator approach we prove a version of the almost sure invariance principle this allows the system to be modelled by a brownian motion up to some error term it follows that many classical statistical properties hold for such systems such as the weak invariance principle and the law of the iterated logarithm

bernoulli coding map and almost sure invariance principle for endomorphisms of mathbb p k let f be an holomorphic endomorphism of mathbb p k and be its measure of maximal entropy we prove an almost sure invariance principle for the systems mathbb p k f mu our class mathcal u of observables includes the holder functions and unbounded ones which present analytic singularities the proof is based on a geometric construction of a bernoulli coding map omega sigma s nu to mathbb p k f mu we obtain the invariance principle for an observable on mathbb p k f mu by applying philipp stout s theorem for chi psi circ omega on s the invariance principle implies the central limit theorem as well as several statistical properties for the class mathcal u as an application we give a direct proof of the absolute continuity of the measure when it satisfies pesin s formula this approach relies on the central limit theorem for the unbounded observable log tt jac f in mathcal u

a hybrid time domain technique that combines the finite element finite difference and method of moment techniques to solve complex electromagnetic problems this paper describes a hybrid technique directly operating in time domain that combines the finite element time domain fetd the finite difference time domain fdtd and the integral equation based method of moments in the time domain momtd techniques to analyze complex electromagnetic problems involving thin wire antennas radiating in the presence of inhomogeneous dielectric bodies whose shape can be arbitrary the method brings together the ability of the fdtd scheme to deal with arbitrary material properties the versatility of the fetd to accurately model curved geometries and that of the mom to analyze thin wire structures working in the time domain provides wide band information from a single execution of the marching on in time procedure and simplifies the interfacing of the fe and mom methods with the fdtd an approach specifically designed for time domain analysis numerical results that validate the hybrid method and show its capabilities are presented in the paper

singularity of projections of 2 dimensional measures invariant under the geodesic flow we show that on any compact riemann surface with variable negative curvature there exists a measure which is invariant and ergodic under the geodesic flow and whose projection to the base manifold is 2 dimensional and singular with respect to the 2 dimensional lebesgue measure

the boltzmann grad limit of the periodic lorentz gas we study the dynamics of a point particle in a periodic array of spherical scatterers and construct a stochastic process that governs the time evolution for random initial data in the limit of low scatterer density boltzmanngrad limit a generic path of the limiting process is a piecewise linear curve whose consecutive segments are generated by a markov process with memory two

a fuzzy sv k modes algorithm for clustering categorical data with set valued attributes abstract in this paper we propose a fuzzy sv k modes algorithm that uses the fuzzy k modes clustering process to cluster categorical data with set valued attributes in the proposed algorithm we use jaccard coefficient to measure the dissimilarity between two objects and represent the center of a cluster with set valued modes a heuristic update way of cluster prototype is developed for the fuzzy partition matrix these extensions make the fuzzy sv k modes algorithm can cluster categorical data with single valued and set valued attributes together and the fuzzy k modes algorithm is its special case experimental results on the synthetic data sets and the three real data sets from different applications have shown the efficiency and effectiveness of the fuzzy sv k modes algorithm

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

superdiffusive limits for deterministic fast slow dynamical systems we consider deterministic fast slow dynamical systems on mathbb r m times y of the form begin aligned left begin array ll x k 1 n x k n n 1 a big x k n big n 1 alpha b big x k n big v y k y k 1 f y k end array right end aligned where alpha in 1 2 under certain assumptions we prove convergence of the m dimensional process x n t x lfloor nt rfloor n to the solution of the stochastic differential equation begin aligned mathrm d x a x mathrm d t b x diamond mathrm d l alpha end aligned where l alpha is an alpha stable levy process and diamond indicates that the stochastic integral is in the marcus sense in addition we show that our assumptions are satisfied for intermittent maps f of pomeau manneville type

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

statistical properties and decay of correlations for interval maps with critical points and singularities we consider a class of piecewise smooth one dimensional maps with critical points and singularities possibly with infinite derivative under mild summability conditions on the growth of the derivative on critical orbits we prove the central limit theorem and a vector valued almost sure invariance principle we also obtain results on decay of correlations and large deviations

non stationary almost sure invariance principle for hyperbolic systems with singularities we investigate a wide class of two dimensional hyperbolic systems with singularities and prove the almost sure invariance principle asip for the random process generated by sequences of dynamically holder observables the observables could be unbounded and the process may be non stationary and need not have linearly growing variances our results apply to anosov diffeomorphisms sinai dispersing billiards and their perturbations the random processes under consideration are related to the fluctuation of lyapunov exponents the shrinking target problem etc

recurrence properties of planar lorentz process first return and first hitting times local times and first intersection times are studied for planar finite horizon lorentz processes with a periodic configuration of scatterers their asymptotic behavior is analogous to the asymptotic behavior of the same quantities for the 2 d simple symmetric random walk cf classical results of darling kac 1957 and of erd s taylor 1960 moreover asymptotical distributions for phases in first hittings and in first intersections of lorentz processes are also described the results are also extended to the quasi one dimensional model of the linear lorentz process subject classification 37d50 billiards 60f05 weak theorems

stochastic partial differential fluid equations as a diffusive limit of deterministic lagrangian multi time dynamics in holm holm 2015 proc r soc a471 20140963 doi 10 1098 rspa 2014 0963 stochastic fluid equations were derived by employing a variational principle with an assumed stochastic lagrangian particle dynamics here we show that the same stochastic lagrangian dynamics naturally arises in a multi scale decomposition of the deterministic lagrangian flow map into a slow large scale mean and a rapidly fluctuating small scale map we employ homogenization theory to derive effective slow stochastic particle dynamics for the resolved mean part thereby obtaining stochastic fluid partial equations in the eulerian formulation to justify the application of rigorous homogenization theory we assume mildly chaotic fast small scale dynamics as well as a centring condition the latter requires that the mean of the fluctuating deviations is small when pulled back to the mean flow

a huygens principle for diffusion and anomalous diffusion in spatially extended systems we present a universal view on diffusive behavior in chaotic spatially extended systems for anisotropic and isotropic media for anisotropic systems strong chaos leads to diffusive behavior brownian motion with drift and weak chaos leads to superdiffusive behavior levy processes with drift for isotropic systems the drift term vanishes and strong chaos again leads to brownian motion we establish the existence of a nonlinear huygens principle for weakly chaotic systems in isotropic media whereby the dynamics behaves diffusively in even space dimension and exhibits superdiffusive behavior in odd space dimensions

almost sure invariance principle for nonuniformly hyperbolic systems we prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems discrete time systems and flows are covered by this result in particular the result applies to the planar periodic lorentz flow with finite horizon statistical limit laws such as the central limit theorem the law of the iterated logarithm and their functional versions are immediate consequences

on almost sure versions of classical limit theorems for dynamical systems the purpose of this article is to support the idea that whenever we can prove a limit theorem in the classical sense for a dynamical system we can prove a suitable almost sure version based on an empirical measure with log average we follow three different approaches martingale methods spectral methods and induction arguments our results apply among others to axiom a maps or flows to systems inducing a gibbs markov map and to the stadium billiard

extreme value theory for non uniformly expanding dynamical systems we establish extreme value statistics for functions with multiple maxima and some degree of regularity on certain non uniformly expanding dynamical systems we also establish extreme value statistics for time series of observations on discrete and continuous suspensions of certain non uniformly expanding dynamical systems via a general lifting theorem the main result is that a broad class of observations on these systems exhibit the same extreme value statistics as i i d processes with the same distribution function

fluctuation of the entropy production for the lorentz gas under small external forces in this paper we study the physical and statistical properties of the periodic lorentz gas with finite horizon driven to a non equilibrium steady state by the combination of non conservative external forces and deterministic thermostats a version of this model was introduced by chernov eyink lebowitz and sinai and subsequently generalized by chernov and the third author non equilibrium steady states for these models are srb measures and they are characterized by the positivity of the steady state entropy production rate our main result is to establish that the entropy production in this context equal to the phase space contraction satisfies the gallavotti cohen fluctuation relation the main tool needed in the proof is the family of anisotropic banach spaces introduced by the first and third authors to study the ergodic and statistical properties of billiards using transfer operator techniques

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

law of iterated logarithm and invariance principle for one parameter families of interval maps we show that for almost every map in a transversal one parameter family of piecewise expanding unimodal maps the birkhoff sum of suitable observables along the forward orbit of the turning point satisfies the law of iterated logarithm this result will follow from an almost sure invariance principle for the birkhoff sum as a function on the parameter space furthermore we obtain a similar result for general one parameter families of piecewise expanding maps on the interval

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

almost sure invariance principle for sequential and non stationary dynamical systems we establish almost sure invariance principles a strong form of approximation by brownian motion for non stationary time series arising as observations on dynamical systems our examples include observations on sequential expanding maps perturbed dynamical systems non stationary sequences of functions on hyperbolic systems as well as applications to the shrinking target problem in expanding systems

homogenization for deterministic maps and multiplicative noise a recent paper of melbourne stuart a note on diffusion limits of chaotic skew product flows nonlinearity 24 2011 1361 1367 gives a rigorous proof of convergence of a fast slow deterministic system to a stochastic differential equation with additive noise in contrast to other approaches the assumptions on the fast flow are very mild in this paper we extend this result from continuous time to discrete time moreover we show how to deal with one dimensional multiplicative noise this raises the issue of how to interpret certain stochastic integrals it is proved that the integrals are of stratonovich type for continuous time and neither

smooth approximation of stochastic differential equations consider an ito process x satisfying the stochastic differential equation dx a x dt b x dw where a b are smooth and w is a multidimensional brownian motion suppose that wn has smooth sample paths and that wn converges weakly to w a central question in stochastic analysis is to understand the limiting behavior of solutions xn to the ordinary differential equation dxn a xn dt b xn dwn the classical wong zakai theorem gives sufficient conditions under which xn converges weakly to x provided that the stochastic integral b x dw is given the stratonovich interpretation the sufficient conditions are automatic in one dimension but in higher dimensions the correct interpretation of b x dw depends sensitively on how the smooth approximation wnis chosen in applications a natural class of smooth approximations arise by setting wn t n 1 2 nt0v sds where t is a flow generated e g by an ordinary differential equation and v is a mean zero observable under mild conditions on t we give a definitive answer to the interpretation question for the stochastic integral b x dw our theory applies to anosov or axiom a flows t as well as to a large class of nonuniformly hyperbolic flows including the one defined by the well known lorenz equations and our main results do not require any mixing assumptions on t the methods used in this paper are a combination of rough path theory and smooth ergodic theory

rapid decay of correlations for nonuniformly hyperbolic flows we show that superpolynomial decay of correlations rapid mixing is prevalent for a class of nonuniformly hyperbolic flows these flows are the continuous time analogue of the class of nonuniformly hyperbolic maps for which young proved exponential decay of correlations the proof combines techniques of dolgopyat and operator renewal theory it follows from our results that planar periodic lorentz flows with finite horizons and flows near homoclinic tangencies are typically rapid mixing

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

decay of correlations and invariance principles for dispersing billiards with cusps and related planar billiard flows following recent work of chernov markarian and zhang it is known that the billiard map for dispersing billiards with zero angle cusps has slow decay of correlations with rate 1 n since the collisions inside a cusp occur in quick succession it is reasonable to expect a much faster decay rate in continuous time in this paper we prove that the flow is rapid mixing correlations decay faster than any polynomial rate a consequence is that the flow admits strong statistical properties such as the almost sure invariance principle even though the billiard map does not the techniques in this paper yield new results for other standard examples in planar billiards including bunimovich flowers and stadia

a fuzzy k modes algorithm for clustering categorical data this correspondence describes extensions to the fuzzy k means algorithm for clustering categorical data by using a simple matching dissimilarity measure for categorical objects and modes instead of means for clusters a new approach is developed which allows the use of the k means paradigm to efficiently cluster large categorical data sets a fuzzy k modes algorithm is presented and the effectiveness of the algorithm is demonstrated with experimental results

a vector valued almost sure invariance principle for sinai billiards with random scatterers understanding the statistical properties of the aperiodic planar lorentz gas stands as a grand challenge in the theory of dynamical systems here we study a greatly simplified but related model proposed by arvind ayyer and popularized by joel lebowitz in which a scatterer configuration on the torus is randomly updated between collisions taking advantage of recent progress in the theory of time dependent billiards on the one hand and in probability theory on the other we prove a vector valued almost sure invariance principle for the model notably the configuration sequence can be weakly dependent and non stationary we provide an expression for the covariance matrix which in the non stationary case differs from the traditional one we also obtain a new invariance principle for sinai billiards the case of fixed scatterers with time dependent observables and improve the accuracy and generality of existing results

the k means type algorithms versus imbalanced data distributions k means is a partitional clustering technique that is well known and widely used for its low computational cost the representative algorithms include the hard k means and the fuzzy k means however the performance of these algorithms tends to be affected by skewed data distributions i e imbalanced data they often produce clusters of relatively uniform sizes even if input data have varied cluster sizes which is called the uniform effect in this paper we analyze the causes of this effect and illustrate that it probably occurs more in the fuzzy k means clustering process than the hard k means clustering process as the fuzzy index m increases the uniform effect becomes evident to prevent the effect of the uniform effect we propose a multicenter clustering algorithm in which multicenters are used to represent each cluster instead of one single center the proposed algorithm consists of the three subalgorithms the fast global fuzzy k means best m plot and grouping multicenter algorithms they will be respectively used to address the three important problems 1 how are the reliable cluster centers from a dataset obtained 2 how are the number of clusters which these obtained cluster centers represent determined 3 how is it judged as to which cluster centers represent the same clusters the experimental studies on both synthetic and real datasets illustrate the effectiveness of the proposed clustering algorithm in clustering balanced and imbalanced data

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

rates in almost sure invariance principle for dynamical systems with some hyperbolicity we prove the almost sure invariance principle with rate o n varepsilon for every varepsilon 0 for holder continuous observables on nonuniformly expanding and nonuniformly hyperbolic transformations with exponential tails examples include gibbs markov maps with big images axiom a diffeomorphisms dispersing billiards and a class of logistic and henon maps the best previously proved rate is o n 1 4 log n 1 2 log log n 1 4 as a part of our method we show that nonuniformly expanding transformations are factors of markov shifts with simple structure and natural metric similar to the classical young towers the factor map is lipschitz continuous and probability measure preserving for this we do not require the exponential tails