Year: 2026
Type: article
Abstract: We investigate local entropy theory, particularly the property of having completely positive entropy (CPE), from a descriptive set-theoretic point of view. We aim to determine the descriptive complexity of different families of dynamical systems with CPE. For a large class of compact metrizable spaces (including manifolds and topological graphs), we show that the family of dynamical systems with CPE is complete coanalytic and hence not Borel. When we restrict our attention to dynamical systems with additional properties, we obtain contrasting behavior. Specifically, CPE and uniformly positive entropy (UPE) – the latter being a Borel property – coincide for mixing maps on topological graphs. However, the family of mixing maps with CPE on Cantor spaces is coanalytic and not Borel. For dynamical systems with the shadowing property, CPE and UPE coincide regardless of the phase space.
Source: Forum of Mathematics Sigma
Authors: Felipe García-Ramos, Udayan B. Darji
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $2,650
OpenAlex ID: W7222919534