[Paper Review] The Ergodic Closing Lemma for Nonsingular Endomorphisms

Armando Castro|ArXiv.org|Jun 11, 2009

Mathematical Dynamics and Fractals12 references3 citations

TL;DR

This paper generalizes Mañé's Ergodic Closing Lemma to $C^1$ nonsingular endomorphisms on compact Riemannian manifolds, proving that for a residual set of such maps, the set of invariant probability measures is the closed convex hull of ergodic measures supported on periodic orbits. The proof combines topological perturbation techniques with ergodic theory and Vitali covering arguments to show that generic orbits are shadowed by periodic orbits in a measure-theoretic sense.

ABSTRACT

We prove the ergodic Closing Lemma for Nonsingular Endomorphisms.

Motivation & Objective

• To extend Mañé's Ergodic Closing Lemma from diffeomorphisms to $C^1$ nonsingular endomorphisms.
• To establish that for a residual subset of nonsingular endomorphisms, invariant probability measures are the closed convex hull of ergodic measures supported on periodic orbits.
• To provide a foundational tool for studying generic hyperbolicity and expansion in endomorphism dynamics.

Proposed method

• Introduces the set $\Sigma(f, \mathcal{U}, \epsilon)$ of points $\epsilon$-shadowed by periodic orbits of nearby maps $g \in \mathcal{U}$.
• Uses a topological perturbation lemma to show that points returning close to themselves can be $\epsilon$-shadowed by periodic orbits of a $C^1$-close endomorphism.
• Applies Birkhoff's Ergodic Theorem and Vitali covering arguments to prove that the shadowing set has full measure for any invariant probability.
• Employs a nested neighborhood basis $\mathcal{U}_n$ to define $\Sigma(f) = \bigcap_n \Sigma(f, \mathcal{U}_n, 1/n)$, ensuring residuality.
• Leverages robustness of hyperbolic periodic points to ensure perturbations preserve hyperbolicity in the residual set.
• Uses weak-* convergence of empirical measures along orbits to show that ergodic measures are limits of periodic orbit measures.

Experimental results

Research questions

• RQ1 Can Mañé's Ergodic Closing Lemma be extended to nonsingular endomorphisms?
• RQ2 Is the set of invariant probability measures for a generic $C^1$ nonsingular endomorphism the closed convex hull of ergodic measures supported on periodic orbits?
• RQ3 Does every ergodic invariant measure for a generic nonsingular endomorphism arise as a weak-* limit of periodic orbit measures?
• RQ4 Can the shadowing of generic orbits by periodic orbits be established in the $C^1$ nonsingular endomorphism setting?
• RQ5 What is the role of residual sets and continuity of the periodic measure hull map in proving generic structural properties?

Key findings

• For a residual subset $\mathcal{R} \subset \mathrm{NEnd}^1(M)$, every $f \in \mathcal{R}$ satisfies that $\mathcal{M}_1(f)$ is the closed convex hull of ergodic measures supported on periodic orbits of $f$.
• The set $\Sigma(f)$ of points $\epsilon$-shadowed by periodic orbits of nearby maps has full measure for any $f$-invariant probability measure.
• For any $f \in \mathrm{NEnd}^1(M)$, the set $\Sigma(f, \mathcal{U}, \epsilon)$ is a total probability set for $f$, proving the non-residual version of the Ergodic Closing Lemma.
• Ergodic measures are weak-* limits of periodic orbit measures from nearby maps $g_k \to f$, with the periodic points hyperbolic and persisting under small perturbations.
• The map $f \mapsto \overline{\mathcal{M}_{\text{per}}(f)}$ is lower semicontinuous, and continuity points of this map form a residual set.
• The residual set $\mathcal{R}$ ensures that the closed convex hull of periodic orbit ergodic measures captures all $f$-ergodic measures, confirming the structural stability conjecture's measure-theoretic aspect.