Computing invariant measures of piecewise convex transformations

LetS:[0, 1][0,1] be a piecewise convex transformation satisfying some conditions which guarantee the existence of an absolutely continuous invariant probability measure. We prove the convergence of a class of Markov finite approximations for computing the invariant measure, using a compactness argument forL ¹-spaces.Research was supported in part by a grant from the Minority Scholars Program through the University of Southern Mississippi.     

Global attractors and invariant measures for non-invertible planar piecewise isometric maps

The authors investigate dynamical behaviors of discrete systems defined by iterating non-invertible planar piecewise isometries, which are piecewisely defined maps that preserve Euclidean distance. After discussing subtleties for these kind of dynamical systems, they have characterized global attractors via invariant measures and via positive continuous functions on phase space. The main result of this Letter is that a compact set A is the global attractor for a piecewise isometry if and only if the Lebesgue measure restricted to A is invariant, while it is not invariant restricted to any measurable set B which contains A and whose Lebesgue measure is strictly larger than that of A.     

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.     

Computing Invariant Densities and Metric Entropy

We present a method for accurately computing the metric entropy (or, equivalently, the Lyapunov exponent) of the absolutely continuous invariant measure μ for a piecewise analytic expanding Markov map T of the interval. We construct atomic signed measures μ _M supported on periodic orbits up to period M, and prove that super-exponentially fast. We illustrate our method with several examples. Received: 25 July 1999 / Accepted: 7 January 2000     

A matrix method for estimating the Liapunov exponent of one-dimensional systems

Let : [0, 1][0, 1] be a piecewise monotonie expanding map. Then admits an absolutely continuous invariant measure. A result of Kosyakin and Sandler shows that can be approximated by a sequence of absolutely continuous measures _n invariant under piecewise linear Markov maps _itn. Each _itn is constructed on the inverse images of the turning points of . The easily computable measures _n are used to estimate the Liapunov exponent of . The idea of using Markov maps for estimating the Liapunov exponent is applied to both expanding and nonexpanding maps.     

Invariant Measures Satisfying an Equality Relating Entropy,Folding Entropy and Negative Lyapunov Exponents

In this paper we prove that, for a C ² (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. This work is supported by National Basic Research Program of China (973 Program) (2007CB814800).     

Constructive approximations to densities invariant under nonexpanding transformations

Let : [0, 1][0, 1] be a piecewise monotonic, nonexpanding map which has an invariant densityg and is topologically conjugate to a piecewise monotonic, expanding map, where the conjugacy is absolutely continuous. An effective, computable method is presented for approximatingg.     

Absolutely Continuous Invariant Measures¶for Piecewise Real-Analytic Expanding Maps¶on the Plane

We prove the existence of absolutely continuous invariant measures for piecewise real-analytic expanding maps on bounded regions in the plane. Received: 5 June 1998 / Accepted: 11 May 1999     

Remark on the (non)convergence of ensemble densities in dynamical systems

We consider a dynamical system with state space M, a smooth, compact subset of some R(n), and evolution given by T(t), x(t)=T(t)x, x in M; T(t) is invertible and the time t may be discrete, t in Z, T(t)=T(t), or continuous, t in R. Here we show that starting with a continuous positive initial probability density rho(x,0)>0, with respect to dx, the smooth volume measure induced on M by Lebesgue measure on R(n), the expectation value of logrho(x,t), with respect to any stationary (i.e., time invariant) measure nu(dx), is linear in t, nu(logrho(x,t))=nu(logrho(x,0))+Kt. K depends only on nu and vanishes when nu is absolutely continuous with respect to dx.(c) 1998 American Institute of Physics.     

Regularity and other properties of absolutely continuous invariant measures for the quadratic family

In the current paper we study in more detail some properties of the absolutely continuous invariant measures constructed in the course of the proof of Jakobson's Theorem. In particular, we show that the density of the invariant measure is continuous at Misiurewicz points. From this we deduce that the Lyapunov exponent is also continuous at these points (our considerations apply just to the parameters constructed in the proof of Jakobson's Theorem). Other properties, like the positivity of the Lyapunov exponent, uniqueness of the absolutely continuous invariant measure and exactness of the corresponding dynamical system, are also proved.This paper was written during the author's stay at the IAS while supported by NSF grant DMS-860 1978     

Marginal singularities,almost invariant sets,and small perturbations of chaotic dynamical systems

For a class of piecewise monotone locally noncontracting maps f:X-->X with small "traps" Y( varepsilon ) subset, dbl equals X (diam(Y( varepsilon )) infinity conditional probabilities that f(n+1)x in X\Y( varepsilon ) if x,fx,.,f(nx) in X\Y( varepsilon ) and the point x is chosen at random. Also proven is the convergence of &mgr;( varepsilon ) to smooth f-invariant measures as varepsilon -->0. By means of this construction, the numerical phenomenon of the convergence of histograms of trajectories of maps with marginal singularities to densities of nonfinite smooth invariant measures in the computer modeling was investigated.     

No or Infinitely Many A.C.I.P.¶for Piecewise Expanding Cr Maps¶in Higher Dimensions

We modify Tsujii's example [9] to show that in contrast with the one-dimensional case, piecewise uniformly expanding and C ^r maps of the plane may: (1) either have no absolutely continuous invariant probability measures (a.c.i.p. for short) and be such that {\bf every point} is statistically attracted to a fixed repelling point;? (2) or have infinitely many ergodic a.c.i.p. Received: 6 September 2000 / Accepted: 15 May 2001     

Multifractal structure of a riddled basin

For a two-dimensional piecewise linear map with a riddled basin, a multifractal spectrum f(gamma), which characterizes the "skeletons" of the riddled basin, is introduced. With f(gamma), the uncertainty exponent is obtained by a variational principle, which enables us to introduce a concept of a "boundary" for the riddled basin. A conjecture on the relation between f(gamma) and the "stable sets" of various ergodic measures, which coexist with the natural invariant measure on the chaotic attractor, is also proposed. (c) 2001 American Institute of Physics.     

Large Deviation Principle for Benedicks-Carleson Quadratic Maps

Since the pioneering works of Jakobson and Benedicks &; Carleson and others, it has been known that a positive measure set of quadratic maps admit invariant probability measures absolutely continuous with respect to Lebesgue. These measures allow one to statistically predict the asymptotic fate of Lebesgue almost every initial condition. Estimating fluctuations of empirical distributions before they settle to equilibrium requires a fairly good control over large parts of the phase space. We use the sub-exponential slow recurrence condition of Benedicks &; Carleson to build induced Markov maps of arbitrarily small scale and associated towers, to which the absolutely continuous measures can be lifted. These various lifts together enable us to obtain a control of recurrence that is sufficient to establish a level 2 large deviation principle, for the absolutely continuous measures. This result encompasses dynamics far from equilibrium, and thus significantly extends presently known local large deviations results for quadratic maps.     

Potentials for almost Markovian random fields

A positive almost Markovian random field is a probability measure on a lattice gas whose finite set conditional probabilities are continuous and positive. We show that each such random field has a potential and in the translation invariant case an absolutely convergent potential. We give a criterion for determining which random fields correspond to pair potentials, or in generaln-body potentials. We show that two translation invariant positive almost Markovian random fields have the same finite set conditional probabilities if and only if one minimizes the specific free energy of the other.     

Measure-theoretical properties of the unstable foliation of two-dimensional differentiable area-preserving systems

This article analyzes in detail the statistical and measure-theoretical properties of the nonuniform stationary measure, referred to as the w-invariant measure, associated with the spatial length distribution of the integral manifolds of the unstable invariant foliation in two-dimensional differentiable area-preserving systems. The analysis is developed starting from a sequence of analytical approximations for the associated density. These approximations are related to the properties of the Jacobian matrix of the nth iteration of a Poincaré map. The w-invariant measure plays a fundamental role in the study of transport phenomena in laminar-chaotic fluid-mixing systems, for which it furnishes the asymptotic invariant distribution of intermaterial contact length between two fluids. The w-invariant measure turns out to be singular and exhibits multifractal features. Its associated density displays local self-similarity in an epsilon neighborhood of hyperbolic periodic points. The cancellation exponent of the signed measure associated with the w measure by attaching at each point the direction of the field of the asymptotic unstable eigenvectors is also analyzed. The only case for which the w-invariant measure is absolutely continuous is given by the conjugation of hyperbolic toral automorphisms with a linear automorphism. The connections with the statistical properties, and in particular with the stretching dynamics, are addressed in detail.     

A new transition for projections of multifractal measures and random maps

Typical projections of simple multifractal measures with generalized dimensionsD _q onto subspaces of dimensionD are considered. It is known that forD _o > D almost all projections have Euclidean support. Here it is shown that if in additionD increases beyondD, a typical projection changes from a singular continuous distribution to an absolutely continuous measure with a squareintegrable, or even differentiable density, and thus from a multifractal to an ordinary distribution with trivial singularity spectrum. Since projections of strictly self-similar measures can be regarded as invariant distributions of iterated function systems, such a transition is found also there and is expected to occur in related systems.     

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.Heisenberg fellow of the Deutsche Forschungsgemeinschaft     

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.     

Rigid rotor as a toy model for Hodge theory

We apply the superfield approach to the toy model of a rigid rotor and show the existence of the nilpotent and absolutely anticommuting Becchi–Rouet–Stora–Tyutin (BRST) and anti-BRST symmetry transformations, under which, the kinetic term and the action remain invariant. Furthermore, we also derive the off-shell nilpotent and absolutely anticommuting (anti-) co-BRST symmetry transformations, under which, the gauge-fixing term and the Lagrangian remain invariant. The anticommutator of the above nilpotent symmetry transformations leads to the derivation of a bosonic symmetry transformation, under which, the ghost terms and the action remain invariant. Together, the above transformations (and their corresponding generators) respect an algebra that turns out to be a physical realization of the algebra obeyed by the de Rham cohomological operators of differential geometry. Thus, our present model is a toy model for the Hodge theory.