Related papers: A measure-theoretic expansion exponent

In this work we analyze the concept of swap-invariance, which is a weaker variant of exchangeability. A random vector $\xi$ in $\mathbb{R}^n$ is called swap-invariant if $\,{\mathbf E}\,\big| \!\sum_j u_j \xi_j \big|\,$ is invariant under…

Given an ergodic measure with positive entropy and only positive Lyapunov exponents, its dynamical quantifiers can be approximated by means of quantifiers of some family of uniformly expanding repellers. Here non-uniformly expanding maps…

We consider random perturbations of a topologically transitive local diffeomorphism of a Riemannian manifold. We show that if an absolutely continuous ergodic stationary measures is expanding (all Lyapunov exponents positive), then there is…

We study homeomorphisms of compact metric spaces whose restriction to the nonwandering set has the pseudo-orbit tracing property. We prove that if there are positively expansive measures, then the topological entropy is positive. Some short…

The aim of this paper is to prove ergodic decomposition theorems for probability measures quasi-invariant under Borel actions of inductively compact groups (Theorem 1) as well as for sigma-finite invariant measures (Corollary 1). For…

The main result of this note, Theorem 2, is the following: a Borel measure on the space of infinite Hermitian matrices, that is invariant under the action of the infinite unitary group and that admits well-defined projections onto the…

The problem of positive Kolmogorov-Sinai entropy of the Chirikov-Standard map with respect to the invariant Lebesgue measure on the two-dimensional is open. In 1999, we believed to have a proof that the entropy can be bounded below. This…

We generate new hierarchy of many-parameter family of maps of the interval [0,1] with an invariant measure, by composition of the chaotic maps of reference [1]. Using the measure, we calculate Kolmogorov-Sinai entropy, or equivalently…

For a class of piecewise hyperbolic maps in two dimensions, we propose a combinatorial definition of topological entropy by counting the maximal, open, connected components of the phase space on which iterates of the map are smooth. We…

We develop the basic theory of ergodic Schr\"odinger operators, which is well known for ergodic probability measures, in the case of a base dynamics on an infinite measure space. This includes the almost sure constancy of the spectrum and…

For a measurable map $T$ and a sequence of $T$-invariant probability measures $\mu_n$ that converges in some sense to a $T$-invariant probability measure $\mu$, an estimate from below for the Kolmogorov--Sinai entropy of $T$ with respect to…

We introduce a family of maps generating continued fractions where the digit $1$ in the numerator is replaced cyclically by some given non-negative integers $(N_1,\ldots,N_m)$. We prove the convergence of the given algorithm, and study the…

We consider a smooth expanding map g on the circle of degree 2. It is known that the Lyapunov exponent of g with respect to the unique invariant measure that is absolutely continuous with respect to the Lebesgue measure is positive and less…

In this paper we introduce a notion of expansiveness for a set valued map defined on a topological space different from that given by Richard Williams at \cite{Wi, Wi2} and prove that the topological entropy of an expansive set valued map…

In this paper, we consider the question of existence and uniqueness of absolutely continuous invariant measures for expanding $C^1$ maps of the circle. This is a question which arises naturally from results which are known in the case of…

For a dynamical system satisfying the approximate product property and asymptotically entropy expansiveness, we characterize a delicate structrue of the space of invariant measures: The ergodic measures of intermediate entropies and…

This work gives a computable formula for the average measure theoretic entropy of a family of expanding on average random Blaschke products, generalizing work by Pujals, Roberts and Shub [Expanding maps of the circle revisited: positive…

Ergodic optimization aims to single out dynamically invariant Borel probability measures which maximize the integral of a given "performance" function. For a continuous self-map of a compact metric space and a dense set of continuous…

We study the family of quadratic maps f_a(x) = 1 - ax^2 on the interval [-1,1] with a between 0 and 2. When small holes are introduced into the system, we prove the existence of an absolutely continuous conditionally invariant measure using…

Expanding Thurston maps were introduced by M. Bonk and D. Meyer with motivation from complex dynamics and Cannon's conjecture from geometric group theory via Sullivan's dictionary. In this paper, we show that the entropy map of an expanding…