Generic Sharp Shift Embedding for Amenable Group Actions with the Uniform Rokhlin Property and Comparison

Abstract

Naryshkin recently proved that a free action of a countably infinite discrete amenable group $G$ on a compact metrizable space $X$ with the uniform Rokhlin property and comparison (URPC) admits an equivariant embedding into $([0,1{]}^m{)}^G$ whenever $\mathrm{mdim}(X,G)<m/2$. We strengthen this result from existence to genericity by a substantially different method: the set of observables $f\in C(X,[0,1{]}^m)$ whose orbit maps are equivariant embeddings into $([0,1{]}^m{)}^G$ is a dense $G_{\delta }$ subset of $C(X,[0,1{]}^m)$. The proof combines an adaptation of the Kakutani skyscraper construction used by Gutman--Tsukamoto with a new polynomial perturbation method, departing from the predominantly linear perturbation approaches used in earlier equivariant embedding literature.

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