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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.16456 |
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Table of 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.