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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.16456 |
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| _version_ | 1866908334054965248 |
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| author | Morales, C. A. |
| author_facet | Morales, C. A. |
| contents | The expansion exponent (or expansion constant) for maps was introduced by Schreiber in \cite{s}. In this paper, we introduce the analogous exponent for measures. We shall prove the following results: The expansion exponent of a measurable maps is equal to the minimum of the expansion exponent taken over the Borel probability measures. In particular, a map expands small distances (in the sense of Reddy \cite{r}) if and only if every Borel probability has positive expansion exponent. Any nonatomic invariant measure with positive expansion exponent is positively expansive in the sense of \cite{m}. For ergodic invariant measures, the Kolmogorov-Sinai entropy is bounded below by the product of the expansion exponent and the measure upper capacity. As a consequence, any ergodic invariant measure with both positive upper capacity and positive expansion exponent must have positive entropy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16456 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A measure-theoretic expansion exponent Morales, C. A. Dynamical Systems 37B25, 37B65 The expansion exponent (or expansion constant) for maps was introduced by Schreiber in \cite{s}. In this paper, we introduce the analogous exponent for measures. We shall prove the following results: The expansion exponent of a measurable maps is equal to the minimum of the expansion exponent taken over the Borel probability measures. In particular, a map expands small distances (in the sense of Reddy \cite{r}) if and only if every Borel probability has positive expansion exponent. Any nonatomic invariant measure with positive expansion exponent is positively expansive in the sense of \cite{m}. For ergodic invariant measures, the Kolmogorov-Sinai entropy is bounded below by the product of the expansion exponent and the measure upper capacity. As a consequence, any ergodic invariant measure with both positive upper capacity and positive expansion exponent must have positive entropy. |
| title | A measure-theoretic expansion exponent |
| topic | Dynamical Systems 37B25, 37B65 |
| url | https://arxiv.org/abs/2504.16456 |