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Deformation and perturbative rigidity near de la Llave examples

Andrey Gogolev, Martin Leguil

TL;DR

The paper addresses the deformation and perturbative rigidity of four-dimensional Anosov diffeomorphisms near de la Llave’s skew-product examples with constant Lyapunov spectra. The authors reduce the problem to suspension flows, construct coarse local coordinates and templates along invariant foliations, and derive precise period-expansion formulas for shadowing periodic orbits. They prove that, generically, isospectral perturbations near de la Llave’s examples must preserve strong stable/unstable foliations, and they show that perturbations can be arranged to force a nonvanishing leading term in these expansions, leading to a $C^{1+\text{H}}$-conjugacy and thus deformation/local rigidity for a dense subset. This work advances the smooth classification near higher-dimensional de la Llave examples and raises questions about global rigidity and volume-preserving analogs, while highlighting a robust perturbative mechanism via cycle expansions and coarse-chart analysis.

Abstract

De la Llave's examples are Anosov diffeomorphisms on the four-torus $\mathbb{T}^4$ with constant Lyapunov spectrum, yet they are not $C^{1}$-conjugate to the linear model or to each other. Nevertheless, we show that such examples are ``locally exceptional'': we prove deformation and local rigidity for generic diffeomorphisms in proximity of de la Llave's examples.

Deformation and perturbative rigidity near de la Llave examples

TL;DR

The paper addresses the deformation and perturbative rigidity of four-dimensional Anosov diffeomorphisms near de la Llave’s skew-product examples with constant Lyapunov spectra. The authors reduce the problem to suspension flows, construct coarse local coordinates and templates along invariant foliations, and derive precise period-expansion formulas for shadowing periodic orbits. They prove that, generically, isospectral perturbations near de la Llave’s examples must preserve strong stable/unstable foliations, and they show that perturbations can be arranged to force a nonvanishing leading term in these expansions, leading to a -conjugacy and thus deformation/local rigidity for a dense subset. This work advances the smooth classification near higher-dimensional de la Llave examples and raises questions about global rigidity and volume-preserving analogs, while highlighting a robust perturbative mechanism via cycle expansions and coarse-chart analysis.

Abstract

De la Llave's examples are Anosov diffeomorphisms on the four-torus with constant Lyapunov spectrum, yet they are not -conjugate to the linear model or to each other. Nevertheless, we show that such examples are ``locally exceptional'': we prove deformation and local rigidity for generic diffeomorphisms in proximity of de la Llave's examples.
Paper Structure (15 sections, 10 theorems, 64 equations, 1 figure)

This paper contains 15 sections, 10 theorems, 64 equations, 1 figure.

Table of Contents

  1. Introduction
  2. de la Llave examples
  3. Deformation and perturbative rigidity
  4. Preparations and structure of the proof
  5. Geometric mechanism of de la Llave examples
  6. Passing to suspension flows
  7. Local coordinates and templates
  8. Coarse local coordinates
  9. Prescribed families of shadowing periodic orbits
  10. The Proof
  11. Asymptotic formula for full Jacobian along shadowing periodic orbits (uniform in the family)
  12. Reinterpreting the leading term of the asymptotic formula in the coarse charts
  13. Non-vanishing of the leading term after a $C^\infty$ perturbation
  14. Arriving at a contradiction via particular choices of a homoclinic point and parameters
  15. On the conjugacy between $5$-dimensional Anosov flows with a fine splitting into $1$-dimensional subbundles

Key Result

Theorem 1.1

Let $F_0$, $G$ be two $C^r$, $r \in(1,\infty]\cup\{\omega\}$, Anosov diffeomorphisms on $\mathbb{T}^2$ which are topologically conjugated, and whose eigenvalues at corresponding periodic points match. Then the conjugacy is $C^{r_*}$ regular, with

Figures (1)

  • Figure 1: Change of sign of the "center" coordinate $\xi_\infty^\circ(s)$ as $s$ changes.

Theorems & Definitions (36)

  • Theorem 1.1: Marco-Moriyón, de la Llave InvIInvIIInvIIIdlLSRB
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Conjecture 1.7
  • Remark 1.8
  • Remark 1.9
  • Proposition 2.4
  • ...and 26 more