2022/02/21 by Lanckriet, Emiel, Szabó, Gábor
#37B05 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2202.10198
For a countable amenable group G and a fixed dimension m≥ 1, we investigate when it is possible to embed a G-space X into the m-dimensional cubical shift ([0,1]m)G. We focus our attention on systems that arise as an extension of an almost finite G-action on a totally disconnected space Y, in the sense of Matui and Kerr. We show that if such a G-space X has mean dimension less than m/2, then X embeds into the (m+1)-dimensional cubical shift. If the distinguished factor G-space Y is assumed to be a subshift of finite type, then this can be improved to an embedding into the m-dimensional cubical shift. This result ought to be viewed as the generalization of a theorem by Gutman-Tsukamoto for G=\mathbb Z to actions of all amenable groups, and represents the first result supporting the Lindenstrauss-Tsukamoto conjecture for actions of groups other than G=ℤk.