Saturation of Generalized Partially Hyperbolic Attractors

TL;DR

This paper proves that generalized partially hyperbolic attractors are saturated and explores their volume properties, showing that generically they have zero volume, but specific modifications can produce positive volume attractors.

Contribution

It establishes the saturation property for generalized partially hyperbolic attractors and constructs examples with positive volume through modifications.

Findings

Generalized partially hyperbolic attractors are saturated.

Most such attractors have zero volume for generic diffeomorphisms.

Modified models can produce attractors with positive volume.

Abstract

We prove the saturation of a generalized partially hyperbolic attractor of a $C^2$ map. As a consequence, we show that any generalized partially hyperbolic horseshoe-like attractor of a $C^1$-generic diffeomorphism has zero volume. In contrast, by modification of Poincar\'e cross section of the geometric model, we build a $C^1$-diffeomorphism with a partially hyperbolic horseshoe-like attractor of positive volume.