[Paper Review] Physical Measures for Certain Partially Hyperbolic Attractors on 3-Manifolds

Ricardo Bortolotti|arXiv (Cornell University)|Jun 2, 2015

Mathematical Dynamics and FractalsMathematics37 references7 citations

TL;DR

This paper establishes the existence and finiteness of physical measures for certain partially hyperbolic attractors on 3-manifolds with neutral central direction, under a transversality condition between unstable leaves projected via the stable foliation. By proving that u-Gibbs states project to absolutely continuous measures under stable projections, the authors show these states coincide with physical measures, whose basins cover full Lebesgue measure. The result extends to robustly nonhyperbolic attractors satisfying the transversality and regularity conditions.

ABSTRACT

In this work, we study ergodic properties of certain partially hyperbolic attractors whose central direction has a neutral behavior, the main feature is a condition of transversality between unstable leaves when projected by the stable holonomy. We prove that partial hyperbolic attractors satisfying conditions of transversality between unstable leaves via the stable holonomy, neutrality in the central direction and regularity of the stable foliation admits a finite number of physical measures, coinciding with the ergodic u-Gibbs States, whose union of the basins has full Lebesgue measure. Moreover, we describe the construction of a family of robustly nonhyperbolic attractors satisfying these properties.

Motivation & Objective

• To establish the existence and finiteness of physical measures for partially hyperbolic attractors on 3-manifolds where the central Lyapunov exponent is zero.
• To extend the theory of physical measures beyond uniformly hyperbolic and non-uniformly hyperbolic systems to the neutral central direction case.
• To provide a geometric condition—transversality of projected unstable leaves via the stable foliation—that ensures physical measures exist and are finite in number.
• To construct robustly nonhyperbolic attractors satisfying the transversality, neutrality, and regularity conditions, demonstrating the existence of such systems.

Proposed method

• Introduces a modified Doeblin-Fortet (Lasota-Yorke) inequality for finite measures on center-unstable manifolds, using an L2-like norm for density regularity.
• Defines a projection operator via the stable foliation to map u-Gibbs states onto center-unstable leaves, leveraging transversality to ensure absolute continuity of the projected measure.
• Uses a contraction mapping argument on Banach spaces of continuous functions to construct the unstable foliation’s H"older regularity, relying on the central direction's neutral behavior.
• Applies a perturbation argument to show that the transversality and regularity conditions are robust under small C1 perturbations of the diffeomorphism.
• Employs a cone field argument to prove that unstable directions remain transverse under perturbations, ensuring the transversality condition persists in an open neighborhood.
• Constructs explicit examples of robustly nonhyperbolic attractors by deforming a base diffeomorphism with neutral central dynamics, ensuring the required geometric and regularity conditions are met.

Experimental results

Research questions

• RQ1 Under what geometric and dynamical conditions do partially hyperbolic attractors with neutral central direction admit finitely many physical measures?
• RQ2 Can the transversality of projected unstable leaves via the stable foliation ensure that u-Gibbs states are physical measures?
• RQ3 Is the existence and finiteness of physical measures robust under small C1 perturbations when the central direction is neutral?
• RQ4 Can one construct robustly nonhyperbolic attractors satisfying the transversality and regularity conditions required for physical measure existence?

Key findings

• Partially hyperbolic attractors satisfying transversality between projected unstable leaves, neutral central direction, and Lipschitz stable foliation admit a finite number of physical measures.
• The physical measures coincide exactly with the ergodic u-Gibbs states, and their basins cover a full Lebesgue measure subset of the attractor’s basin of attraction.
• The transversality condition ensures that the local stable projection of u-Gibbs states yields absolutely continuous measures on center-unstable leaves, which is key to identifying them as physical measures.
• Robustly nonhyperbolic attractors satisfying the three conditions (transversality, neutrality, stable foliation regularity) exist and can be explicitly constructed.
• The stable foliation is C1-continuous in a neighborhood of the base diffeomorphism, ensuring the results hold for an open set of systems.
• The attractor is robustly transitive and nonhyperbolic due to the presence of periodic points of different indices, confirming the robustness of the construction.