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

TL;DR

This paper investigates the structure of hyperbolic components, called tongues, in the Double Standard Map family, providing a uniformization, characterizing maximal chaotic sets, and analyzing their Hausdorff dimension variation.

Contribution

It introduces a real-analytic uniformization for each tongue, characterizes the maximal chaotic set, and studies the Hausdorff dimension variation within tongues.

Findings

Proved simple connectedness of tongues.

Characterized the unique maximal chaotic subset.

Showed Hausdorff dimension varies real-analytically inside a tongue.

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.