Journal•ISSN: 0143-3857
Ergodic Theory and Dynamical Systems
Cambridge University Press
About: Ergodic Theory and Dynamical Systems is an academic journal published by Cambridge University Press. The journal publishes majorly in the area(s): Ergodic theory & Invariant (mathematics). It has an ISSN identifier of 0143-3857. Over the lifetime, 3485 publications have been published receiving 83063 citations. The journal is also known as: Ergodic theory & dynamical systems.
Topics: Ergodic theory, Invariant (mathematics), Mathematics, Measure (mathematics), Dynamical systems theory
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TL;DR: In this article, the authors consider diffeomorphisms of surfaces leaving invariant an ergodic Borel probability measure and define HD (μ) to be the infimum of Hausdorff dimension of sets having full μ-measure.
Abstract: We consider diffeomorphisms of surfaces leaving invariant an ergodic Borel probability measure μ. Define HD (μ) to be the infimum of Hausdorff dimension of sets having full μ-measure. We prove a formula relating HD (μ) to the entropy and Lyapunov exponents of the map. Other classical notions of fractional dimension such as capacity and Renyi dimension are discussed. They are shown to be equal to Hausdorff dimension in the present context.
634 citations
TL;DR: In this article, it was shown that for any amenable non-singular countable equivalence relation R⊂X×X, there exists a nonsingular transformation T of X such that, up to a null set, any two Cartan subalgebras of a hyperfinite factor are conjugate by an automorphism.
Abstract: We prove that for any amenable non-singular countable equivalence relation R⊂X×X, there exists a non-singular transformation T of X such that, up to a null set:It follows that any two Cartan subalgebras of a hyperfinite factor are conjugate by an automorphism
591 citations
TL;DR: In this article, it was shown that when the Julia set J of a rational function f is hyperbolic, the Hausdorff dimension of J depends real analytically on f.
Abstract: The purpose of this note is to prove a conjecture of D. Sullivan that when the Julia set J of a rational function f is hyperbolic, the Hausdorff dimension of J depends real analytically on f. We shall obtain this as corollary of a general result on repellers of real analytic maps (see corollary 5).
463 citations
TL;DR: This method essentially gives the optimal polynomial bound for the decay of correlations, the degree depending on the order of the tangency at the neutral fixed point.
Abstract: We present an original approach which allows us to investigate the statistical properties of a non-uniformly hyperbolic map on the interval. Based on a stochastic approximation of the deterministic map, this method essentially gives the optimal polynomial bound for the decay of correlations, the degree depending on the order of the tangency at the neutral fixed point.
435 citations
TL;DR: In this article, the existence of a unique measure of maximal entropy for rational endomorphisms of the Riemann sphere is established and the equidistribution of pre-images and periodic points with respect to this measure is proved.
Abstract: In this paper the existence of a unique measure of maximal entropy for rational endomorphisms of the Riemann sphere is established. The equidistribution of pre-images and periodic points with respect to this measure is proved.
430 citations