Uniformization of tongues in Double Standard Map family and variation of maximal chaotic sets
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912360927592448 |
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| author | Banerjee, Kuntal Bhattacharyya, Anubrato Mukherjee, Sabyasachi |
| author_facet | Banerjee, Kuntal Bhattacharyya, Anubrato Mukherjee, Sabyasachi |
| contents | We study hyperbolic components, also known as tongues, in the Double Standard Map family comprising circle maps of the form:
\begin{align*}
f_{a,b}(x)=\left(2x+a+\dfrac{b}π \sin(2πx)\right) \mod 1,\ a \in \mathbb{R}/\mathbb{Z},\ 0 \leq b \leq 1. \end{align*} We prove simple connectedness of tongues by providing a dynamically natural real-analytic uniformization for each tongue. For maps in a tongue, we characterize the unique maximal subset of the circle on which $f_{a,b}$ is Devaney chaotic. We also show that the Hausdorff dimension of this maximal chaotic set varies real-analytically inside a tongue. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02189 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniformization of tongues in Double Standard Map family and variation of maximal chaotic sets Banerjee, Kuntal Bhattacharyya, Anubrato Mukherjee, Sabyasachi Dynamical Systems 30D05, 37D05, , 37D35, 37E10, 37F10, 37F15, 37F31, 37F35 (Primary), 37A30, 37B10, 37D20 (Secondary) We study hyperbolic components, also known as tongues, in the Double Standard Map family comprising circle maps of the form: \begin{align*} f_{a,b}(x)=\left(2x+a+\dfrac{b}π \sin(2πx)\right) \mod 1,\ a \in \mathbb{R}/\mathbb{Z},\ 0 \leq b \leq 1. \end{align*} We prove simple connectedness of tongues by providing a dynamically natural real-analytic uniformization for each tongue. For maps in a tongue, we characterize the unique maximal subset of the circle on which $f_{a,b}$ is Devaney chaotic. We also show that the Hausdorff dimension of this maximal chaotic set varies real-analytically inside a tongue. |
| title | Uniformization of tongues in Double Standard Map family and variation of maximal chaotic sets |
| topic | Dynamical Systems 30D05, 37D05, , 37D35, 37E10, 37F10, 37F15, 37F31, 37F35 (Primary), 37A30, 37B10, 37D20 (Secondary) |
| url | https://arxiv.org/abs/2505.02189 |