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

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.