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On the entropy of processes generated by quasifactors

Rômulo M. Vermersch

Abstract

Given a measurable dynamical system $(X,\mathcal{X},μ,T)$, where $X$ is a compact metric space, $\mathcal{X}$ is the Borel $σ$-algebra on $X$, $μ$ is a $T$-invariant Borel probability measure and $T$ is a homeomorphism acting on $X$ we show that, if $h_μ(T)>0$, then $h_{\widetildeμ}(\widetilde{T})>0$ for every quasifactor $\widetildeμ$ of $μ$ having full-support.

On the entropy of processes generated by quasifactors

Abstract

Given a measurable dynamical system , where is a compact metric space, is the Borel -algebra on , is a -invariant Borel probability measure and is a homeomorphism acting on we show that, if , then for every quasifactor of having full-support.

Paper Structure

This paper contains 3 sections, 7 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Our results

Key Result

Proposition 1

Let $(X,\mathcal{X},\mu,T)$ be a MDS, $0<\mu(A)<1$, $\mu(\partial A)=0$, $0<\eta<1$ and let $\widetilde{\mu}\in Q(\mu)$ having full-support. If $\widetilde{A}:=\{\nu\in\mathcal{M}(X):\nu(A)>\eta\}$, then $0<\widetilde{\mu}(\widetilde{A})<1$.

Theorems & Definitions (15)

  • Proposition 1
  • proof
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Corollary 6
  • ...and 5 more