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PreprintarXiv licencearXiv

Entropy Scales in Topological Dynamical Systems

  • Yunxiang Xie
  • Ercai Chen
  • Xiaoyao Zhou
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Abstract

In this paper, we define topological and measure-theoretic entropy scales in dynamical systems for the scaling introduced by Helfter. Firstly, we establish a variational principle between Bowen and lower entropy scales on compact subsets. Secondly, we prove factor inequalities for Bowen and upper capacity entropy scales on arbitrary subsets; unlike the classical additive case, the fiber contribution is described by a transfer function and need not be additive. We also study entropy scales and sequence entropy scales under the induced action. Under suitable conditions, zero values are preserved in both directions, while positive values become infinite; counterexamples show that additional assumptions are generally necessary. As an application, we use entropy scales to distinguish zero-entropy systems sharing the same entropy dimension but differ in finer orbit-growth rates.

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Cite

BibTeXRIS
BibTeX
@misc{xie2026entropy,
  title = {{Entropy Scales in Topological Dynamical Systems}},
  author = {Yunxiang Xie and Ercai Chen and Xiaoyao Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.10460},
  eprint = {2608.10460},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.10460},
}
RIS
TY  - UNPB
TI  - Entropy Scales in Topological Dynamical Systems
AU  - Yunxiang Xie
AU  - Ercai Chen
AU  - Xiaoyao Zhou
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2608.10460
UR  - https://arxiv.org/abs/2608.10460
AB  - In this paper, we define topological and measure-theoretic entropy scales in dynamical systems for the scaling introduced by Helfter. Firstly, we establish a variational principle between Bowen and lower entropy scales on compact subsets. Secondly, we prove factor inequalities for Bowen and upper capacity entropy scales on arbitrary subsets; unlike the classical additive case, the fiber contribution is described by a transfer function and need not be additive. We also study entropy scales and sequence entropy scales under the induced action. Under suitable conditions, zero values are preserved in both directions, while positive values become infinite; counterexamples show that additional assumptions are generally necessary. As an application, we use entropy scales to distinguish zero-entropy systems sharing the same entropy dimension but differ in finer orbit-growth rates.
ER  -