[Paper Review] A hyperbolic diffeomorphism with countably many ergodic components near identity

Huyi Hu, Anna Talitskaya|ArXiv.org|Oct 19, 2006

Mathematical Dynamics and Fractals6 references3 citations

TL;DR

This paper constructs a smooth, volume-preserving, hyperbolic diffeomorphism on a four-dimensional compact Riemannian manifold that is arbitrarily close to the identity map and has countably many ergodic components. Using perturbations of a geodesic flow on a surface of constant negative curvature, the authors remove two zero Lyapunov exponents while preserving ergodicity and hyperbolicity, demonstrating that such systems exist in arbitrarily small neighborhoods of the identity.

ABSTRACT

We construct a smooth hyperbolic volume preserving diffeomorphism on a four dimensional compact Riemannian manifold which has countably many ergodic components and is arbitrarily close to the identity map.

Motivation & Objective

• To construct a smooth, volume-preserving, hyperbolic diffeomorphism on a 4-manifold that is arbitrarily close to the identity map.
• To demonstrate the existence of such systems with countably many ergodic components, resolving a gap in the understanding of dynamical systems near identity.
• To overcome the challenge of removing two zero Lyapunov exponents while preserving hyperbolicity and ergodicity in a four-dimensional setting.
• To extend previous results on non-uniformly hyperbolic systems with infinitely many ergodic components to systems arbitrarily close to the identity.

Proposed method

• Perturb a product of a geodesic flow on a surface of constant negative curvature and the identity on a circle to construct the base map F = G × id.
• Partition the circle S¹ into countably many disjoint intervals In, and define local perturbations fn on M₀ × In using affine conjugation via πn.
• Apply three small C∞ volume-preserving perturbations to the base map S = G × id to achieve ergodicity and remove zero Lyapunov exponents.
• Use a rotation-like map Tθ on local coordinates to create a perturbation that preserves volume and has small C¹ norm, ensuring closeness to identity.
• Ensure the perturbation preserves the derivative structure at the endpoints of each interval In to maintain smoothness and C¹ closeness to identity.
• Leverage accessibility via symbolic dynamics and periodic orbit density to construct sets where accessibility holds almost everywhere, enabling ergodicity.

Experimental results

Research questions

• RQ1 Can a C∞ hyperbolic volume-preserving diffeomorphism with countably many ergodic components be constructed arbitrarily close to the identity map?
• RQ2 How can two zero Lyapunov exponents be removed in a four-dimensional system without destabilizing the non-zero exponents from prior perturbations?
• RQ3 Is it possible to achieve ergodicity in a four-dimensional system with two zero Lyapunov exponents using perturbations that preserve volume and C¹ closeness to identity?
• RQ4 Does the existence of such systems near identity contradict the non-existence of Anosov systems near identity?

Key findings

• A C∞ diffeomorphism f on a 4-dimensional compact Riemannian manifold exists such that ||f − id||C₁ ≤ δ₀ for any δ₀ > 0.
• The map f preserves the Riemannian volume μ and is hyperbolic, with all Lyapunov exponents non-zero almost everywhere.
• The system has countably many ergodic components, each open modulo zero, confirming the existence of such systems near identity.
• The construction uses three perturbations: one to achieve accessibility, one to remove the first zero Lyapunov exponent, and a third to remove the second while preserving hyperbolicity.
• The perturbation technique ensures that the derivative structure is preserved at the boundary of each interval In, maintaining C¹ closeness to the identity.
• The result shows that non-uniformly hyperbolic systems with infinitely many ergodic components can exist arbitrarily close to the identity, contrasting with the non-existence of Anosov systems near identity.