Mean Dimension & Jaworski-type Theorems

TL;DR

This paper extends embedding theorems for finite and infinite dimensional dynamical systems using mean dimension and periodic points, confirming a conjecture and introducing new marker concepts.

Contribution

It proves a conjecture by Lindenstrauss and Tsukamoto on embedding finite dimensional systems with periodic points into higher-dimensional cubical shifts.

Findings

Finite dimensional aperiodic systems embed in 1D cubical shift.

Systems with periodic points and bounded periodic dimension embed in higher-dimensional shifts.

Infinite dimensional systems with similar periodic dimension conditions can have their periodic points immersed in shifts.

Abstract

According to the celebrated Jaworski Theorem, a finite dimensional aperiodic dynamical system $(X,T)$ embeds in the $1$-dimensional cubical shift $([0,1]^{\mathbb{Z}},shift)$. If $X$ admits periodic points (still assuming $\dim(X)<\infty$) then we show in this paper that periodic dimension $perdim(X,T)<\frac{d}{2}$ implies that $(X,T)$ embeds in the $d$-dimensional cubical shift $(([0,1]^{d})^{\mathbb{Z}},shift)$. This verifies a conjecture by Lindenstrauss and Tsukamoto for finite dimensional systems. Moreover for an infinite dimensional dynamical system, with the same periodic dimension assumption, the set of periodic points can be equivariantly immersed in $(([0,1]^{d})^{\mathbb{Z}},shift)$. Furthermore we introduce a notion of markers for general topological dynamical systems, and use a generalized version of the Bonatti-Crovisier tower theorem, to show that an extension $(X,T)$ of an aperiodic finite-dimensional system whose mean dimension obeys $mdim(X,T)<\frac{d}{16}$ embeds in the $(d+1)$-cubical shift $(([0,1]^{d+1})^{\mathbb{Z}},shift)$.