[Paper Review] On the existence of non-hyperbolic ergodic measures as the limit of periodic measures

Christian Bonatti, Jinhua Zhang|arXiv (Cornell University)|Jun 20, 2016

Mathematical Dynamics and Fractals15 references3 citations

TL;DR

This paper establishes the existence of non-hyperbolic ergodic measures as weak* limits of hyperbolic periodic measures in two robust dynamical settings: robustly transitive diffeomorphisms far from homoclinic tangencies and diffeomorphisms with robust cycles. Using a shadowing lemma from [G], the authors construct periodic orbits with vanishing center Lyapunov exponents without perturbing the dynamics, proving that such non-hyperbolic measures—supported on the full manifold or compact invariant sets—arise as limits of periodic orbits.

ABSTRACT

[GIKN] and [BBD1] propose two very different ways for building non hyperbolic measures, [GIKN] building such a measure as the limit of periodic measures and [BBD1] as the $ω$-limit set of a single orbit, with a uniformly vanishing Lyapunov exponent. The technique in [GIKN] was essentially used in a generic setting, as the periodic orbits were built by small perturbations. It is not known if the measures obtained by the technique in [BBD1] are accumulated by periodic measures. In this paper we use a shadowing lemma from [G]: $\bullet$for getting the periodic orbits in [GIKN] without perturbing the dynamics, $\bullet$for recovering the compact set in [BBD1] with a uniformly vanishing Lyapunov exponent by considering the limit of periodic orbits. As a consequence, we prove that there exists an open and dense subset $\mathcal{U}$ of the set of robustly transitive non-hyperbolic diffeomorphisms far from homoclinic tangencies, such that for any $f\in\mathcal{U}$, there exists a non-hyperbolic ergodic measure with full support and approximated by hyperbolic periodic measures. We also prove that there exists an open and dense subset $\mathcal{V}$ of the set of diffeomorphisms exhibiting a robust cycle, such that for any $f\in\mathcal{V}$, there exists a non-hyperbolic ergodic measure approximated by hyperbolic periodic measures.

Motivation & Objective

• To resolve the open question of whether non-hyperbolic ergodic measures constructed via the [BBD1] method (as ω-limit sets with vanishing Lyapunov exponents) can be approximated by periodic measures.
• To show that the non-hyperbolic ergodic measures built via the [GIKN] criterion (as weak* limits of periodic measures) can be realized without perturbing the dynamics.
• To unify two distinct constructions of non-hyperbolic measures by proving both can be realized as limits of hyperbolic periodic measures under open and dense geometric conditions.
• To establish that such non-hyperbolic ergodic measures have full support and are approximated by periodic orbits with vanishing center Lyapunov exponents.

Proposed method

• Apply a shadowing lemma from [G] to construct periodic orbits in the neighborhood of a robust cycle without performing dynamical perturbations.
• Use the [GIKN] criterion (via [BDG, Lemma 2.5]) to ensure that weak* limits of periodic measures are ergodic and non-hyperbolic.
• Construct a sequence of hyperbolic periodic orbits whose center Lyapunov exponents decay exponentially to zero, ensuring the limit measure is non-hyperbolic.
• Employ the 'controlled at any scale' criterion from [BBD1] to recover the compact invariant set with vanishing center Lyapunov exponent as the Hausdorff limit of periodic orbits.
• Use uniform continuity of the Lyapunov exponent functions and uniform size of stable/unstable manifolds to ensure homoclinic relations between periodic orbits.
• Apply Lemma 2.3 to guarantee convergence of Dirac measures on periodic orbits to a non-hyperbolic ergodic measure with full support.

Experimental results

Research questions

• RQ1 Can non-hyperbolic ergodic measures constructed via the ω-limit set method in [BBD1] be approximated by periodic measures?
• RQ2 Can the periodic orbit construction in [GIKN] be realized without perturbing the dynamics, using shadowing techniques?
• RQ3 Is there an open and dense subset of robustly transitive non-hyperbolic diffeomorphisms where non-hyperbolic ergodic measures with full support arise as weak* limits of periodic measures?
• RQ4 Can the compact invariant set with vanishing center Lyapunov exponent in [BBD1] be recovered as the Hausdorff limit of periodic orbits?
• RQ5 Do the weak* limits of periodic measures supported on orbits with vanishing center Lyapunov exponents yield ergodic measures supported on the same compact set?

Key findings

• There exists an open and dense subset 𝒰 of robustly transitive non-hyperbolic diffeomorphisms far from homoclinic tangencies such that for each f ∈ 𝒰, there exists a non-hyperbolic ergodic measure with full support that is the weak* limit of hyperbolic periodic measures.
• For any f in an open and dense subset 𝒱 of diffeomorphisms with a robust cycle, there exists a non-hyperbolic ergodic measure that is the weak* limit of hyperbolic periodic measures.
• The sequence of periodic orbits constructed via the shadowing lemma has center Lyapunov exponents decaying exponentially to zero, ensuring the limit measure is non-hyperbolic.
• The Hausdorff limit of the periodic orbits contains the compact invariant set K′_f from [BBD1], which supports the non-hyperbolic ergodic measure with vanishing center Lyapunov exponent.
• Any weak* limit measure of the periodic measures is either supported on K′_f or is a unique periodic measure, implying the limit measure is ergodic and non-hyperbolic.
• The support of the limiting non-hyperbolic ergodic measure is the entire manifold M, due to the ε_n-dense approximation of periodic orbits with ε_n → 0.