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arXiv · 2505.02189

Uniformization of tongues in Double Standard Map family and variation of maximal chaotic sets

Kuntal Banerjee·Anubrato Bhattacharyya·Sabyasachi Mukherjee
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Abstract

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}{\pi} \sin(2\pi x)\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.

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Kuntal Banerjee, Anubrato Bhattacharyya, Sabyasachi Mukherjee. 2025-05-04. Uniformization of tongues in Double Standard Map family and variation of maximal chaotic sets. https://doi.org/10.1088/1361-6544%2Fae8a05

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