Expansive properties of induced dynamical systems
| Autor: | Daniel Jardón, Iván Sánchez |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
0209 industrial biotechnology
Pure mathematics Dynamical systems theory Logic 02 engineering and technology Metric space 020901 industrial engineering & automation Compact space Artificial Intelligence Metric (mathematics) 0202 electrical engineering electronic engineering information engineering 020201 artificial intelligence & image processing Locally compact space Dynamical system (definition) Expansive Mathematics |
| Zdroj: | Fuzzy Sets and Systems. 425:48-61 |
| ISSN: | 0165-0114 |
| Popis: | For a given metric space X, we consider the set of all normal fuzzy sets on X, denoted by F ( X ) . In this work, we study expanding, positively expansive and weakly positively expansive dynamical systems ( X , f ) and how they are reflected in the dynamical system ( F ( X ) , f ˆ ) , where f ˆ is the Zadeh's extension of f and F ( X ) has one of the following metrics: the levelwise metric, the endograph metric, the sendograph metric and the Skorokhod metric. We mainly show that if we consider the following conditions: (i) ( X , f ) is positively expansive (resp. expanding); (ii) ( K ( X ) , f ‾ ) is positively expansive (resp. expanding); (iii) ( F ∞ ( X ) , f ˆ ) is positively expansive (resp. expanding); (iv) ( F 0 ( X ) , f ˆ ) is positively expansive (resp. expanding). Then (iv)⇒ (iii) ⇔ (ii) ⇒ (i). For expanding dynamical systems, we present a compact metric space and a locally compact metric space to show that (i) ⇏ (ii) and (iii) ⇏ (iv), respectively. For positively expansive dynamical systems, there is a compact metric space satisfying that (i) ⇏ (ii), but we don't know if (iii) ⇒ (iv). |
| Databáze: | OpenAIRE |
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