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Non-stationary normal coordinates on neighborhoods of Pesin stable manifolds

Masato Tsujii

TL;DR

This work develops non-stationary normal coordinates around Pesin stable manifolds by extending the GK/KKS non-stationary normal-form framework to a neighborhood that includes normal directions, using jets to control transversal behavior. By formulating a non-autonomous dynamics on $ bR^d$ with a Lyapunov-decomposed linear part and introducing a jet-equivalence calculus, the authors prove that, after appropriate coordinate changes, the $oldsymbol{ extell}$-jets of the dynamics lie in a finite-dimensional Lie group of sub-resonance generated polynomial automorphisms, with the nonlinearity uniquely determined up to this group. In the Pesin setting, this yields Lyapunov-adapted coordinates in which the transversal dynamics are governed by the same finite-dimensional structure and the coordinate changes grow sub-exponentially, enabling a geometric description of stable and unstable foliations through jet data. The result provides a robust framework for fractal geometric analysis of foliations and suggests applications to rigidity and statistical properties (e.g., mixing) in smooth dynamical systems, including potential use in analyzing holonomy regularity via jet-level descriptions.

Abstract

We construct non-stationary normal coordinates in a neighborhood of Pesin stable manifolds \cite{Pesin}. This construction is a natural extension, via jets in the normal directions, of the non-stationary normal coordinates on stable manifolds introduced by Guysinsky and Katok. We emphasize that this extension provides a useful framework for describing the fractal geometric structures of stable and unstable foliations in smooth dynamical systems.

Non-stationary normal coordinates on neighborhoods of Pesin stable manifolds

TL;DR

This work develops non-stationary normal coordinates around Pesin stable manifolds by extending the GK/KKS non-stationary normal-form framework to a neighborhood that includes normal directions, using jets to control transversal behavior. By formulating a non-autonomous dynamics on with a Lyapunov-decomposed linear part and introducing a jet-equivalence calculus, the authors prove that, after appropriate coordinate changes, the -jets of the dynamics lie in a finite-dimensional Lie group of sub-resonance generated polynomial automorphisms, with the nonlinearity uniquely determined up to this group. In the Pesin setting, this yields Lyapunov-adapted coordinates in which the transversal dynamics are governed by the same finite-dimensional structure and the coordinate changes grow sub-exponentially, enabling a geometric description of stable and unstable foliations through jet data. The result provides a robust framework for fractal geometric analysis of foliations and suggests applications to rigidity and statistical properties (e.g., mixing) in smooth dynamical systems, including potential use in analyzing holonomy regularity via jet-level descriptions.

Abstract

We construct non-stationary normal coordinates in a neighborhood of Pesin stable manifolds \cite{Pesin}. This construction is a natural extension, via jets in the normal directions, of the non-stationary normal coordinates on stable manifolds introduced by Guysinsky and Katok. We emphasize that this extension provides a useful framework for describing the fractal geometric structures of stable and unstable foliations in smooth dynamical systems.
Paper Structure (12 sections, 3 theorems, 52 equations, 2 figures)

This paper contains 12 sections, 3 theorems, 52 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Non-autonomous dynamics on $\mathbb{R}^d$
  3. Settings
  4. Coordinates and Jets
  5. Sub-resonance generated polynomial automorphisms
  6. Main Result
  7. An application to the framework of Pesin theory
  8. Discussion
  9. Proofs
  10. Proof of Lemma \ref{['lm:SR']}
  11. Proof of Theorem \ref{['th:abs']}
  12. Proof of Theorem \ref{['th:mainX']}

Key Result

Lemma 2.2

The set forms a finite-dimensional Lie group with respect to composition. Similarly, but slightly differently, the set is closed under composition. (The inverse of an element of $\mathcal{G}^{(\ell)}_-$ does not exist as a polynomial map in general.)

Figures (2)

  • Figure 1: The action of $\mathcal{L}_{\boldsymbol{\chi}}$ around $0$.
  • Figure 2: Local stable manifolds for points on $W^u_{loc}(p)$.

Theorems & Definitions (7)

  • Remark 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 3.1
  • Remark 3.2
  • Remark 5.1