Dynamical Embedding in Cubical Shifts & the Topological Rokhlin and Small Boundary Properties

TL;DR

This paper investigates conditions under which topological dynamical systems can be embedded into cubical shifts, introducing local markers and establishing their equivalence to the Rokhlin property, with implications for mean dimension and boundary properties.

Contribution

It verifies the Lindenstrauss-Tsukamoto conjecture for systems with finite-dimensional non-wandering sets and introduces local markers as a new tool for analyzing embedding properties.

Findings

Marker property is equivalent to a topological Rokhlin Lemma.

Extensions of aperiodic systems with countably many minimal subsystems have the marker property.

Vanishing mean dimension is equivalent to the small boundary property for systems with the marker property.

Abstract

According to a conjecture of Lindenstrauss and Tsukamoto, a topological dynamical system $(X,T)$ is embeddable in the $d$-cubical shift $(([0,1]^{d})^{\mathbb{Z}},\ shift)$ if both its mean dimension and periodic dimension are strictly bounded by $\frac{d}{2}$. We verify the conjecture for the class of systems admitting finite dimensional non-wandering sets (under the additional assumption of closed periodic points set). The main tool in the proof is the new concept of local markers. Continuing the investigation of (global) markers initiated in previous work it is shown that the marker property is equivalent to a topological version of the Rokhlin Lemma. Moreover new classes of systems are found to have the marker property, in particular, extensions of aperiodic systems with a countable number of minimal subsystems. Extending work of Lindenstrauss we show that for systems with the marker property vanishing mean dimension is equivalent to the small boundary property. Finally we answer a question by Downarowicz in the affirmative: the small boundary property is equivalent to admitting a zero-dimensional isomorphic extension.