On the entropy of processes generated by quasifactors

TL;DR

This paper proves that positive entropy in a dynamical system implies positive entropy in all its full-support quasifactors, extending the understanding of entropy transfer in such systems.

Contribution

It establishes that positive entropy in a system guarantees positive entropy in all its full-support quasifactors, a novel insight in dynamical systems theory.

Findings

Positive entropy in the original system implies positive entropy in all full-support quasifactors.

The result applies to systems with a homeomorphism on a compact metric space.

The theorem extends the understanding of entropy behavior under quasifactors.

Abstract

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