[Paper Review] Hyperbolicity versus non-hyperbolic ergodic measures inside homoclinic classes

Cheng, Cheng, Sylvain Crovisier+3 more|arXiv (Cornell University)|Jul 29, 2015

Mathematical Dynamics and Fractals17 references3 citations

TL;DR

This paper proves that for $C^1$-generic diffeomorphisms, any non-hyperbolic homoclinic class necessarily supports a non-hyperbolic ergodic measure, confirming a conjecture by Díaz and Gorodetski. The argument relies on analyzing Lyapunov exponents and perturbations to detect heterodimensional cycles or periodic sinks when hyperbolicity fails, using genericity to rule out periodic points of different index and establish measure non-hyperbolicity.

ABSTRACT

We prove that, for $C^1$-generic diffeomorphisms, if a homoclinic class is not hyperbolic, then there is a non-hyperbolic ergodic measure supported on it. This proves a conjecture by Díaz and Gorodetski [28]. We also discuss the conjectured existence of periodic points with different stable dimension in the class.

Motivation & Objective

• To resolve a conjecture by Díaz and Gorodetski on the existence of non-hyperbolic ergodic measures in non-hyperbolic homoclinic classes of $C^1$-generic diffeomorphisms.
• To establish a local version of the broader conjecture on non-hyperbolic dynamics by focusing on homoclinic classes rather than global systems.
• To clarify the dynamical implications of non-hyperbolicity in homoclinic classes by analyzing Lyapunov exponents and periodic orbit structures.
• To demonstrate that the absence of hyperbolicity in a homoclinic class implies the existence of a non-hyperbolic invariant measure, using perturbation techniques and genericity arguments.

Proposed method

• Utilizes $C^1$-genericity to ensure that periodic points are dense in chain-recurrent sets and that homoclinic classes coincide with chain-recurrence classes containing periodic orbits.
• Applies the $C^1$-stability conjecture and results on non-uniform hyperbolicity to analyze Lyapunov exponents of invariant measures.
• Employs perturbation techniques to detect heterodimensional cycles or sinks when Lyapunov exponents approach zero, using Theorem 2.9 and Proposition 3.1.
• Analyzes dominated splittings and finest dominated splittings over the homoclinic class to classify possible dynamical structures.
• Uses the fact that non-hyperbolicity implies either a non-hyperbolic measure or the existence of periodic points with different indices, and rules out the latter via genericity.
• Applies results on approximation of measures by periodic orbits and the behavior of central Lyapunov exponents under small perturbations.

Experimental results

Research questions

• RQ1 Does every non-hyperbolic homoclinic class of a $C^1$-generic diffeomorphism support a non-hyperbolic ergodic measure?
• RQ2 Can the absence of hyperbolicity in a homoclinic class be detected through the behavior of Lyapunov exponents of periodic orbits and their measures?
• RQ3 What dynamical obstructions arise when a homoclinic class contains periodic points with different stable dimensions?
• RQ4 Under what conditions can a perturbation generate a heterodimensional cycle or a sink in a non-hyperbolic homoclinic class?
• RQ5 Is it possible to rule out the existence of periodic points with different indices in a non-hyperbolic homoclinic class under genericity assumptions?

Key findings

• For $C^1$-generic diffeomorphisms, if a homoclinic class $H(p)$ is not hyperbolic, then it supports at least one non-hyperbolic ergodic measure.
• The existence of a sequence of periodic points in $H(p)$ with Lyapunov exponents approaching zero from either side implies the possibility of generating a heterodimensional cycle via $C^1$-small perturbations.
• If all periodic points in $H(p)$ have the same index and the class is non-hyperbolic, then either a partially hyperbolic splitting with a one-dimensional center bundle exists, or the class admits a finest dominated splitting with a two-dimensional center bundle.
• When the central Lyapunov exponent of periodic orbits approaches zero from the positive side, arbitrarily small perturbations can create a sequence of sinks converging to $H(p)$, implying non-hyperbolicity of the measure.
• The non-existence of periodic points with different indices in $H(p)$ is guaranteed under $C^1$-genericity, which allows the conclusion that non-hyperbolicity must be reflected in the measure-theoretic structure.
• The result does not extend to aperiodic classes, as there exist $C^1$-generic diffeomorphisms with aperiodic classes that only support hyperbolic ergodic measures.