Mean dimension and a non-embeddable example for amenable group actions

Abstract

For every infinite (countable discrete) amenable group G and every positive integer d we construct a minimal G-action of mean dimension d/2 which cannot be embedded in the full G-shift on ([0, 1] d ) G .

The Hilbert cube contains a minimal subshift of full mean dimension

“…To construct the minimal subshift, we need to borrow some useful results which have been proved in like [5,4] and also been restated in [9]. Lemma 2.4.…”

Mean Li–yorke Chaotic Set With Full Hausdorff Dimension for Continued Fractions

“…Since then, research into the mean dimension of actions by amenable groups has garnered significant attention. Recently, Jin, Park, and Qiao [9] considered embedding problem for amenable group actions, they devised a minimal system with a mean dimension of d/2 that cannot be embedded in ([0, 1] d ) G for any positive integer d. Based on the tiling property of amenable groups, Shi and Zhang [20] proposed that the induced transformation on the set of probability measures endowed with the weak * topology exhibits infinite topological mean dimension if and only if the system for countably infinite discrete amenable group actions has positive topological entropy, Chen, Dou, and Zheng [2] investigate the variational principle between metric mean dimension and rate-distortion functions for countably infinite amenable group actions, extending the earlier work of Lindenstrauss and Tsukamoto [15] from Z-actions to amenable group actions. Li [13] introduced the concepts of upper measure-theoretic mean dimensions and upper metric mean dimensions for amenable group actions, establishing a variational principle between them.…”