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Peer-reviewedCC BY 4.0Forum of Mathematics, Sigma

Local entropy theory and descriptive complexity

  • Felipe García-Ramos1,2
  • Udayan B. Darji3
  1. 1Universidad Nacional Autónoma de México
  2. 2Jagiellonian University
  3. 3Charles University
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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.

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Cite

BibTeXRIS
BibTeX
@article{garciaramos2026local,
  title = {{Local entropy theory and descriptive complexity}},
  author = {Felipe García-Ramos and Udayan B. Darji},
  journal = {Forum of Mathematics, Sigma},
  year = {2026},
  month = oct,
  doi = {10.1017/fms.2026.10277},
  url = {https://www.cambridge.org/core/product/identifier/S2050509426102771/type/journal_article},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Local entropy theory and descriptive complexity
AU  - Felipe García-Ramos
AU  - Udayan B. Darji
PY  - 2026
DA  - 2026/10/19
JO  - Forum of Mathematics, Sigma
DO  - 10.1017/fms.2026.10277
UR  - https://www.cambridge.org/core/product/identifier/S2050509426102771/type/journal_article
AB  - 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.
SN  - 2050-5094
ER  -