Flexibility versus genericity of phase diagrams of perturbed continuous maps on the Cantor set
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913969575297024 |
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| author | Marsan, Hugo Sablik, Mathieu |
| author_facet | Marsan, Hugo Sablik, Mathieu |
| contents | Consider the dynamical system constitued by a continuous function $F:\mathcal{A}^\mathbb{N}\to\mathcal{A}^\mathbb{N}$ where $\mathcal{A}$ is a finite alphabet. The perturbed counterpart, denoted by $F_ε$, is obtained after each iteration of $F$ by modifying each cell independently with probability $ε\in[0,1]$ and choosing the new value uniformly. We characterize the possible sets of $ε\in[0,1]$ such that $F_ε$ has a unique measure. These sets are exactly the $G_δ$ sets (countable intersection of open sets) of $[0, 1]$ which contain 1. However, we show that generically this set is $]0, 1]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_00461 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Flexibility versus genericity of phase diagrams of perturbed continuous maps on the Cantor set Marsan, Hugo Sablik, Mathieu Dynamical Systems 37A50, 37A05, 60K35, 60J05 Consider the dynamical system constitued by a continuous function $F:\mathcal{A}^\mathbb{N}\to\mathcal{A}^\mathbb{N}$ where $\mathcal{A}$ is a finite alphabet. The perturbed counterpart, denoted by $F_ε$, is obtained after each iteration of $F$ by modifying each cell independently with probability $ε\in[0,1]$ and choosing the new value uniformly. We characterize the possible sets of $ε\in[0,1]$ such that $F_ε$ has a unique measure. These sets are exactly the $G_δ$ sets (countable intersection of open sets) of $[0, 1]$ which contain 1. However, we show that generically this set is $]0, 1]$. |
| title | Flexibility versus genericity of phase diagrams of perturbed continuous maps on the Cantor set |
| topic | Dynamical Systems 37A50, 37A05, 60K35, 60J05 |
| url | https://arxiv.org/abs/2508.00461 |