Flexibility of Entropies for Piecewise Expanding Unimodal Maps
We investigate the flexibility of the entropy (topological and metric) for the class of piecewise expanding unimodal maps. We show that the only restrictions for the values of the topological and metric entropies in this class are that both are positive, the topological entropy is at most log 2, and...
| Autores: | , , |
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| Tipo de recurso: | capítulo de libro |
| Fecha de publicación: | 2024 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:303684 |
| Acceso en línea: | https://ddd.uab.cat/record/303684 https://dx.doi.org/urn:doi:10.1017/9781009278898.003 |
| Access Level: | acceso abierto |
| Palabra clave: | Flexibility Unimodal maps Skew tent maps Piecewise expanding maps Topological entropy Metric entropy |
| Sumario: | We investigate the flexibility of the entropy (topological and metric) for the class of piecewise expanding unimodal maps. We show that the only restrictions for the values of the topological and metric entropies in this class are that both are positive, the topological entropy is at most log 2, and the metric entropy is not larger than the topological entropy. In order to have a better control on the metric entropy, we work mainly with topologically mixing piecewise expanding skew tent maps, for which there are only 2 different slopes. For those maps, there is an additional restriction that the topological entropy is larger than 1/2 log 2. Moreover, we generalize and give a different interpretation of the Milnor-Thurston formula connecting the topological entropy and the kneading determinant for unimodal maps. |
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