Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Kuntal Banerjee, Anubrato Bhattacharyya, Sabyasachi Mukherjee

TL;DR

The paper analyzes tongues (hyperbolic components) in the complexified Double Standard Map family g_{a,b}(z)=e^{2\pi i a} z^2 \exp(b z - b/z) and establishes a dynamically natural, real-analytic uniformization that proves each tongue interior is simply connected, via a diffeomorphism to the disk with a slit. It introduces a new invariant, the critical angle, alongside the real multiplier to parameterize tongues, and proves a real-analytic dependence of the maximal chaotic set on tongue parameters. Inside tongues, the circle dynamics on S^1 has a maximal Devaney chaotic set C_{a,b} which is a Cantor set of Lebesgue measure zero, equal to the Julia set intersection with S^1, and is shown to vary analytically in Hausdorff dimension through thermodynamic formalism and conformal repeller techniques. Outside the hyperbolic locus, the circle map is chaotic on all of S^1, while within tongues chaos is confined to C_{a,b}; the authors develop a holomorphic extension, Markov partitions, and pressure theory to code the dynamics on C_{a,b} and prove the Hausdorff dimension t_{a,b} is the unique zero of P(σ, t·\tilde{ψ}_{a,b}), with t_{a,b} varying real- analytically with (a,b). Altogether, the work provides a complete analytic description of Tongue geometry and the parameter dependence of maximal chaotic sets in the complexified DSM family, bridging quasiconformal deformation, conformal dynamics, and thermodynamic formalism.

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}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.

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

TL;DR

The paper analyzes tongues (hyperbolic components) in the complexified Double Standard Map family g_{a,b}(z)=e^{2\pi i a} z^2 \exp(b z - b/z) and establishes a dynamically natural, real-analytic uniformization that proves each tongue interior is simply connected, via a diffeomorphism to the disk with a slit. It introduces a new invariant, the critical angle, alongside the real multiplier to parameterize tongues, and proves a real-analytic dependence of the maximal chaotic set on tongue parameters. Inside tongues, the circle dynamics on S^1 has a maximal Devaney chaotic set C_{a,b} which is a Cantor set of Lebesgue measure zero, equal to the Julia set intersection with S^1, and is shown to vary analytically in Hausdorff dimension through thermodynamic formalism and conformal repeller techniques. Outside the hyperbolic locus, the circle map is chaotic on all of S^1, while within tongues chaos is confined to C_{a,b}; the authors develop a holomorphic extension, Markov partitions, and pressure theory to code the dynamics on C_{a,b} and prove the Hausdorff dimension t_{a,b} is the unique zero of P(σ, t·\tilde{ψ}_{a,b}), with t_{a,b} varying real- analytically with (a,b). Altogether, the work provides a complete analytic description of Tongue geometry and the parameter dependence of maximal chaotic sets in the complexified DSM family, bridging quasiconformal deformation, conformal dynamics, and thermodynamic formalism.

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}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 is Devaney chaotic. We also show that the Hausdorff dimension of this maximal chaotic set varies real-analytically inside a tongue.
Paper Structure (16 sections, 29 theorems, 42 equations, 3 figures)

This paper contains 16 sections, 29 theorems, 42 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background on the complexified DSM family
  3. Basic properties
  4. Connectedness of tongues à la Dezotti
  5. Dynamically natural uniformization of tongues
  6. Introducing a new conformal invariant
  7. A model change of coordinates in the linearized planes
  8. Uniformization of tongues
  9. Parameter dependence of maximal chaotic set
  10. Maximal set of Devaney chaos
  11. Parameters outside hyperbolic closure
  12. Parameters in tongues
  13. Conformal repeller and symbolic dynamics of the maximal chaotic set
  14. Conformal repeller
  15. Coding, distortion estimates, and pressure
  16. ...and 1 more sections

Key Result

Theorem A

The interior of each tongue in the complexified DSM family is simply connected. Specifically, there is a real-analytic, dynamically natural diffeomorphism from the interior of each tongue onto $\mathbb{D}\setminus[0,1)$. In particular, any two maps in the interior of a tongue are quasiconformally co

Figures (3)

  • Figure 1: Depicted are tongues of period $\leq 10$ in the parameter space, $\mathbb{R/Z}\times [0,1]$, of the DSM family. Here $(a,b) \in [-1/2,1/2] \times [0,1]$
  • Figure 2: Depicted is a schematic diagram of the quasiconformal deformation of Lemma \ref{['qc_def_lemma']}. The red points on the left are the critical points of $g_0, g_1$, and those on the right are the images of the critical points under the corresponding linearizing maps.
  • Figure 3: Displayed are the possible configurations of the intervals $I_i$ and $J_i$. The red intervals are $I_i$, while $J_i=(l_i,r_i)$.

Theorems & Definitions (58)

  • Theorem A
  • Theorem B
  • Lemma 2.1
  • proof
  • Definition 2.2: Tongues in the complexified DSM family
  • Lemma 2.3
  • Definition 3.1: Critical angle
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • ...and 48 more