Pluripotency of wandering dynamics
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909155592241152 |
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| author | Kiriki, Shin Nakano, Yushi Soma, Teruhiko |
| author_facet | Kiriki, Shin Nakano, Yushi Soma, Teruhiko |
| contents | This paper proposes a new concept of pluripotency inspired by Colli-Vargas [Ergod. Theory Dyn. Syst., 21(6):1657-1681, 2001] and presents fundamental theorems for developing the theory. Pluripotency reprograms dynamics from a statistical or geometrical point of view. This means that the dynamics of various codes, including non-trivial Dirac physical measures or historic behavior, can be observably and stochastically realized by arbitrarily small perturbations. We first give a practical condition equivalent to a stronger version of pluripotency. Next, we show that the property of pluripotency is $C^{r} (2\leq r<\infty)$-robust. Precisely, there exists a $C^{r}$-open set of non-hyperbolic diffeomorphisms that have wild blender-horseshoes and are strongly pluripotent. It implies a new affirmative solution to Takens' last problem for $C^{r}$ diffeomorphisms of dimension $n\geq 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_00337 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pluripotency of wandering dynamics Kiriki, Shin Nakano, Yushi Soma, Teruhiko Dynamical Systems Primary: 37C20, 37C29, 37C70, Secondary: 37C25 This paper proposes a new concept of pluripotency inspired by Colli-Vargas [Ergod. Theory Dyn. Syst., 21(6):1657-1681, 2001] and presents fundamental theorems for developing the theory. Pluripotency reprograms dynamics from a statistical or geometrical point of view. This means that the dynamics of various codes, including non-trivial Dirac physical measures or historic behavior, can be observably and stochastically realized by arbitrarily small perturbations. We first give a practical condition equivalent to a stronger version of pluripotency. Next, we show that the property of pluripotency is $C^{r} (2\leq r<\infty)$-robust. Precisely, there exists a $C^{r}$-open set of non-hyperbolic diffeomorphisms that have wild blender-horseshoes and are strongly pluripotent. It implies a new affirmative solution to Takens' last problem for $C^{r}$ diffeomorphisms of dimension $n\geq 3$. |
| title | Pluripotency of wandering dynamics |
| topic | Dynamical Systems Primary: 37C20, 37C29, 37C70, Secondary: 37C25 |
| url | https://arxiv.org/abs/2404.00337 |