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A Proof of the Persistence of Anti-integrable States for Three-Dimensional Quadratic Diffeomorphisms

Yi-Chiuan Chen

TL;DR

This work proves that for a 3D quadratic diffeomorphism in AI limit regimes, AI states that are hyperbolic persist to genuine orbits of the map $L$ as $ε>0$ grows, via an implicit-function-theorem approach. A compact, backward-and-forward complete hyperbolic AI-set $oldsymbol{Λ}$ for the associated quadratic relation yields a compact invariant set $oldsymbol{ ext{A}}_e$ for $L$, with a topological conjugacy between AI sequences and orbits, under a uniform bound on the inverse of the linearization. The results extend recent contraction-map insights by showing the bijection between AI states and orbits can be strengthened to a conjugacy and that the continuation forms a hyperbolic horseshoe, providing Cantor-set dynamics for the 3D map. Applications to HM2022/HM2024 AI states demonstrate both single-branch and two-branch quadratic correspondences, producing robust hyperbolic invariant sets and illustrating the method’s reach beyond purely algebraic AI limits. The discussion also addresses infinite Jacobian limits, alternative invertibility proofs, and the equivalence of hyperbolicity and nondegeneracy, outlining future work on degenerate AI limits.

Abstract

Three-dimensional quadratic diffeomorphisms with quadratic inverse generically have five independent parameters. When some parameters approach infinity, the diffeomorphisms may exhibit a so-called anti-integrable limit in the traditional sense of Aubry and Abramovici. That is, the dynamics of the diffeomorphisms reduce to symbolic dynamics on a finite number of symbols. However, the diffeomorphisms may reduce to quadratic correspondences when parameters approach infinity, and the traditional anti-integrable limit does not deal with this situation. Meiss asked what about an anti-integrable limit for it. Remarkable progress was achieved very recently by the work of Hampton and Meiss [SIAM J. Appl. Dyn. Syst. 21 (2022), pp. 650--675]. Using the contraction mapping theorem, they showed there is a bijection between the anti-integrable states and the sequences of branches of a quadratic correspondence. They also showed that an anti-integrable state can be continued to a genuine orbit of the three-dimensional diffeomorphism. This paper aims to contribute to the progress, by means of the implicit function theorem. We shall show that, under a slightly more restricted condition than that imposed by Hampton and Meiss, the bijection indeed is a topological conjugacy and establish the uniform hyperbolicity of the continued genuine orbits.

A Proof of the Persistence of Anti-integrable States for Three-Dimensional Quadratic Diffeomorphisms

TL;DR

This work proves that for a 3D quadratic diffeomorphism in AI limit regimes, AI states that are hyperbolic persist to genuine orbits of the map as grows, via an implicit-function-theorem approach. A compact, backward-and-forward complete hyperbolic AI-set for the associated quadratic relation yields a compact invariant set for , with a topological conjugacy between AI sequences and orbits, under a uniform bound on the inverse of the linearization. The results extend recent contraction-map insights by showing the bijection between AI states and orbits can be strengthened to a conjugacy and that the continuation forms a hyperbolic horseshoe, providing Cantor-set dynamics for the 3D map. Applications to HM2022/HM2024 AI states demonstrate both single-branch and two-branch quadratic correspondences, producing robust hyperbolic invariant sets and illustrating the method’s reach beyond purely algebraic AI limits. The discussion also addresses infinite Jacobian limits, alternative invertibility proofs, and the equivalence of hyperbolicity and nondegeneracy, outlining future work on degenerate AI limits.

Abstract

Three-dimensional quadratic diffeomorphisms with quadratic inverse generically have five independent parameters. When some parameters approach infinity, the diffeomorphisms may exhibit a so-called anti-integrable limit in the traditional sense of Aubry and Abramovici. That is, the dynamics of the diffeomorphisms reduce to symbolic dynamics on a finite number of symbols. However, the diffeomorphisms may reduce to quadratic correspondences when parameters approach infinity, and the traditional anti-integrable limit does not deal with this situation. Meiss asked what about an anti-integrable limit for it. Remarkable progress was achieved very recently by the work of Hampton and Meiss [SIAM J. Appl. Dyn. Syst. 21 (2022), pp. 650--675]. Using the contraction mapping theorem, they showed there is a bijection between the anti-integrable states and the sequences of branches of a quadratic correspondence. They also showed that an anti-integrable state can be continued to a genuine orbit of the three-dimensional diffeomorphism. This paper aims to contribute to the progress, by means of the implicit function theorem. We shall show that, under a slightly more restricted condition than that imposed by Hampton and Meiss, the bijection indeed is a topological conjugacy and establish the uniform hyperbolicity of the continued genuine orbits.
Paper Structure (15 sections, 10 theorems, 110 equations, 1 figure)

This paper contains 15 sections, 10 theorems, 110 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Quadratic correspondences and AI states
  3. Persistence of AI states
  4. Invertibility of $D\mathcal{L}(\boldsymbol\xi; e)$
  5. Proof of Theorem \ref{['mainthm']}
  6. Applications
  7. A deterministic one-dimensional map
  8. A pair of one-dimensional maps
  9. Discussions
  10. Infinite Jacobian determinant: $\delta_1\not=0$
  11. Invertibility of $D\mathcal{L}(\boldsymbol\xi; e)$ revisited
  12. Alternative proof of Lemma \ref{['mainla']} (with a different uniform bound).
  13. Hyperbolicity and nondegeneracy
  14. Degenerate AI limit
  15. Conclusions

Key Result

Proposition 2.1

A hyperbolic compact set for the relation $\mathcal{E}$ is expansive.

Figures (1)

  • Figure 1: A schematic illustration of $\mathcal{E}(0,1, 1-\bar{c}, \bar{c})$ for $0.8<\bar{c}<1$: The relation is depicted in black, and the diagonals $v=\pm u$ are dashed blue. The set $\Lambda$ is a Cantor set and equals $\bigcap_{n=0}^\infty f_+^{-n}([0,1/\bar{c}]\setminus (u_L, u_R))$, where $\{u_L, u_R\}$ is the pre-image of $1/\bar{c}$ under $f_+$.

Theorems & Definitions (15)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem 3.1: Main theorem
  • Lemma 3.2
  • Proposition 4.1
  • Proposition 4.2
  • Remark 4.3
  • Remark 4.4
  • Corollary 4.5
  • Remark 5.1
  • ...and 5 more