AG Lie-Gruppen: G. Szabó, KU Leuven: Embedding Topological Dynamics over Amenable Groups into Cubical Shifts

Embedding Topological Dynamics over Amenable Groups into Cubical Shifts – Vortragender: Gábor Szabó, Katholieke Universiteit Leuven – Einladender: Kang Li

Abstract: Given any homeomorphism $T: X\to X$ on a compact metrizable space, it is not hard to see that the topological dynamical system $(X,T)$ embeds into the Hilbert cube $\big( [0,1]^{\mathbb{N}}\big)^{\mathbb{Z}}$ equivariantly with the index shift. It is a much more subtle question (even when $T$ is assumed to be minimal) whether one can always embed into $\big( [0,1]^d \big)^{\mathbb{Z}}$, for some natural number $d$ or even $d=1$, with the index shift. The systematic negative answer for this question was given by Lindenstrauss and Weiss as an application of their concept of mean dimension. Shortly afterwards, Lindenstrauss proved that small enough mean dimension leads to the existence of such embeddings. This phenomenon eventually gave rise to the Lindenstrauss-Tsukamoto conjecture, which is a direct dynamical analog of the Menger-Nöbeling theorem for topological spaces. Groundbreaking advances have taken place for the case of single homeomorphisms, but the entire problem makes perfect sense for topological actions of other groups. In this talk, I will talk about a partial verification of the Lindenstrauss-Tsukamoto conjecture that is applicable to actions of countable amenable groups. This is based on joint work with Emiel Lanckriet.