2023/03/22 by Tomasz Downarowicz, Downarowicz, Tomasz, Mateusz Więcek +1
Mathematics · #37A35 #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Primary: 37B40 #Secondary: 43A07 #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2303.12923
openalex publication_date 2023/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We provide a definition of a \prec-asymptotic pair in a topological action of a countable group G, where \prec is an order on G of type \mathbb Z. We then prove that if G is a countable amenable group and (X,G) is a topological G-action of positive entropy, then for every multiorder (\mathcal O,ν,G) and ν-almost every order \prec ∈\mathcal O there exists a \prec-asympotic pair in X. This result is a generalization of the Blanchard-Host-Ruette Theorem for classical topological dynamical systems (actions of~\mathbb Z). We also prove that for every countable amenable group G, and every multiorder on G arising from a tiling system, every topological G-action of entropy zero has an extension which has no \prec-asymptotic pairs for any \prec belonging to this multiorder. Together, these two theorems give a characterization of topological G-actions of entropy zero: (X,G) has topological entropy zero if and only if, for any multiorder \mathcal O_\boldsymbol\mathsf T on G arising from a tiling system of entropy zero, there exists an extension (Y,G) of (X,G), which has no \prec-asymptotic pairs for any \prec ∈\mathcal O_\boldsymbol\mathsf T, equivalently, there exists a multiorder (\mathcal O,ν,G) on G, such that for ν-almost any \prec ∈\mathcal O, there are no \prec-asymptotic pairs in (Y,G).