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On the Density of Periodic Measures for Star Vector Fields

Qimai Sun, Guangwa Wang, Wanlou Wu

TL;DR

The paper proves that for a $C^1$ star vector field $X$ on a compact $d$-manifold, every ergodic hyperbolic invariant measure not supported on singularities can be approximated in the weak$^*$ topology by periodic measures. The authors develop Lyapunov-norm techniques and Lyapunov charts in the presence of singularities, leveraging the scaled linear Poincaré flow $\psi_t^*$ and its dominated splitting to obtain a shadowing framework (Liao shadowing) for quasi-hyperbolic strings. They then perform a case analysis on the Lyapunov spectrum: if all exponents share the same sign, the measure is supported on a periodic orbit; if not, a dominated splitting coincides with the Oseledets splitting, enabling the construction of a periodic orbit that approximates the given ergodic measure in the weak$^*$ sense. This extends Katok’s density result from $C^{1+\alpha}$ diffeomorphisms to $C^1$ star vector fields, addressing singularities and higher-dimensional settings.

Abstract

In this paper, we prove that every ergodic hyperbolic invariant measure of a $C^1$ star vector field can be approximated by periodic measures in weak$^*$ topology. This extends a classical result of Katok \cite{Ka} for $C^{1+α}(α>0)$ diffeomorphisms to $C^1$ star vector field of any dimension.

On the Density of Periodic Measures for Star Vector Fields

TL;DR

The paper proves that for a star vector field on a compact -manifold, every ergodic hyperbolic invariant measure not supported on singularities can be approximated in the weak topology by periodic measures. The authors develop Lyapunov-norm techniques and Lyapunov charts in the presence of singularities, leveraging the scaled linear Poincaré flow and its dominated splitting to obtain a shadowing framework (Liao shadowing) for quasi-hyperbolic strings. They then perform a case analysis on the Lyapunov spectrum: if all exponents share the same sign, the measure is supported on a periodic orbit; if not, a dominated splitting coincides with the Oseledets splitting, enabling the construction of a periodic orbit that approximates the given ergodic measure in the weak sense. This extends Katok’s density result from diffeomorphisms to star vector fields, addressing singularities and higher-dimensional settings.

Abstract

In this paper, we prove that every ergodic hyperbolic invariant measure of a star vector field can be approximated by periodic measures in weak topology. This extends a classical result of Katok \cite{Ka} for diffeomorphisms to star vector field of any dimension.
Paper Structure (8 sections, 1 theorem, 106 equations)

This paper contains 8 sections, 1 theorem, 106 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic Contents of Vector Fields
  4. The Lyapunov Exponents of Vector Fields
  5. Lyapunov Norms and Shadowing Lemma for Vector Fields
  6. Lyapunov Norms and Lyapunov Charts
  7. Shadowing lemma
  8. Density of periodic measures: proof of Main Theorem

Key Result

Theorem 3.5

Assume that $X\in\mathfrak{X}^1(M)$. Let $\Lambda\subset M\backslash\text{Sing}(X)$ be an invariant set admitting a dominated splitting $\mathcal{N}_\Lambda=E\oplus F$ with respect to the scaled linear Poincaré flow. Given $\eta>0$ and $T>0$. For $\alpha>0$ and $\varepsilon>0$, there exist a constan there exist a strictly increasing $C^1$ function $\theta:[0,T_0]\to\mathbb{R}$ and a periodic point

Theorems & Definitions (5)

  • Remark
  • Remark
  • Theorem 3.5
  • proof : Proof of Main Theorem
  • proof : Proof of Claim