arXiv:2505.02189v2

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

Kuntal Banerjee, Anubrato Bhattacharyya, Sabyasachi Mukherjee

math.DS30D0537D0537D3537E1037F1037F1537F3137F3537A3037B1037D20

Paper metadata

Original submission: May 4, 2025
Latest revision: July 25, 2026 · version 2
Claimed MSC: 30D05, 37D05, 37D35, 37E10, 37F10, 37F15, 37F31, 37F35, 37A30, 37B10, 37D20
arXiv categories: math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: Final accepted version
Journal reference: Nonlinearity, vol. 29, no. 7, Paper no. 075025, 2026
DOI: 10.1088/1361-6544/ae8a05 ↗

Abstract

We study hyperbolic components, also known as tongues, in the Double Standard Map family comprising circle maps of the form: $\begin{aligned} f_{a,b}(x)=\left(2x+a+\dfrac{b}\pi \sin(2\pi x)\right) \mod 1,\ a \in \mathbb{R}/\mathbb{Z},\ 0 \leq b \leq 1. \end{aligned}$ 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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Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsCorrect

The two advertised results are supported. Theorem A/3.5 gives the global multiplier–critical-angle uniformization of each tongue interior and analytic quasiconformal conjugacies. Corollary 1.1/Theorem 4.20 identifies the maximal chaotic set as a moving expanding Cantor repeller and proves that its Hausdorff dimension varies real analytically.

Theorem A / Theorem 3.5Correct

Dynamically natural uniformization of tongue interiors

Pages 4–6 · Theorem A and Corollary 1.1 · arXiv:2505.02189v2

The multiplier and critical-angle coordinates define a real-analytic diffeomorphism from the interior of each tongue in the complexified double-standard-map family onto the slit disk. This proves simple connectivity and gives quasiconformal conjugacies that depend real analytically on the parameter. The attracting cycle is unique in the tongue, and both off-circle critical points lie in the specified immediate basin component, so the critical angle is defined throughout the domain. Boundary values corresponding to the slit are excluded exactly as stated.

Corollary 1.1 / Theorem 4.20Correct

Real-analytic variation of chaotic-set dimension

Pages 3–4 and 24–25 · Corollary 1.1 / Theorem 4.20 · arXiv:2505.02189v2

Inside a tongue, the complement of the unique attracting basin is a compact expanding Cantor set and is the maximal invariant set on which the circle map is Devaney chaotic. The uniformization supplies a real-analytic motion and a fixed Markov coding. The pressure zero varies analytically because its derivative with respect to dimension is strictly negative, so Bowen's formula yields real-analytic Hausdorff dimension throughout the tongue.

Propositions 4.2–4.15Correct

Identification of the maximal chaotic set

Pages 18–23 · Propositions 4.2–4.15 · arXiv:2505.02189v2

Outside hyperbolic tongues the whole circle has the stated chaotic behavior; inside a tongue the basin complement $C_{a,b}$ is closed, forward invariant, topologically transitive, and contains dense periodic points. Expansion away from the attracting basin makes it a Cantor set of zero Lebesgue measure. Any orbit with chaotic recurrence must avoid the attracting basin, establishing maximality in the sense used by Corollary 1.1.

02Proofs3 reported findingsCorrect

Multiplier/critical-angle coordinates and Bowen's formula. The multiplier and critical-angle map is locally real analytic because the attracting basin and its two off-circle critical points move holomorphically. Quasiconformal surgery realizes every target coordinate in the slit disk, normalized rigidity gives injectivity, and properness at the tongue boundary upgrades the local coordinate to a global diffeomorphism. The Beltrami coefficients are chosen equivariantly, so the resulting conjugacies respect the dynamics and depend analytically on the real parameters. These steps establish all parts of Theorem A, not only simple connectivity.

Quasiconformal deformation and pressureCorrect and complete

Multiplier/critical-angle coordinates and Bowen's formula

Pages 15–53 · Sections 3–4 · arXiv:2505.02189v2

The multiplier and critical-angle map is locally real analytic because the attracting basin and its two off-circle critical points move holomorphically. Quasiconformal surgery realizes every target coordinate in the slit disk, normalized rigidity gives injectivity, and properness at the tongue boundary upgrades the local coordinate to a global diffeomorphism. The Beltrami coefficients are chosen equivariantly, so the resulting conjugacies respect the dynamics and depend analytically on the real parameters. These steps establish all parts of Theorem A, not only simple connectivity.

Sections 3.1–3.3Correct and complete

Multiplier–angle coordinates and quasiconformal surgery

Pages 8–18 · Sections 3.1–3.3 · arXiv:2505.02189v2

Linearization in the immediate attracting basin defines the critical angle independently of the chosen coordinate normalization. The surgery changes multiplier and angle while preserving the symmetry and degree of the double-standard family. Solving the invariant Beltrami equation returns a normalized map in that family. Rigidity shows equal coordinates give conformally conjugate normalized maps and hence equal parameters; degeneration analysis gives properness at every boundary approach.

Section 4Correct and complete

Pressure zero and analytic implicit-function argument

Pages 18–25 · Section 4 · arXiv:2505.02189v2

A finite Markov partition codes $C_{a,b}$ by one subshift throughout a tongue. Transfer-operator perturbation gives real-analytic pressure $P(a,b,t)$ , and expansion makes $\partial P/\partial t$ the negative integral of $\log|g'|$ , hence nonzero. Bowen's formula identifies the unique root with Hausdorff dimension. The analytic implicit-function theorem therefore applies at every interior parameter, and overlapping local branches agree by uniqueness.

03Novelty0 reported findingsNo non-novelty findings

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Paper: arXiv:2505.02189v2
Authors listed: Kuntal Banerjee, Anubrato Bhattacharyya, Sabyasachi Mukherjee
Audit date: August 18, 2026

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

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Dynamically natural uniformization of tongue interiors

Real-analytic variation of chaotic-set dimension

Identification of the maximal chaotic set

Multiplier/critical-angle coordinates and Bowen's formula

Multiplier–angle coordinates and quasiconformal surgery

Pressure zero and analytic implicit-function argument

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