Accessibility and central integrability in the absence of periodic points

TL;DR

This paper investigates the accessibility properties of certain partially hyperbolic diffeomorphisms on 3-manifolds without periodic points, revealing conditions for accessibility and integrability of the center bundle.

Contribution

It establishes new conditions under which such diffeomorphisms are accessible and characterizes the structure of accessibility classes in these systems.

Findings

Accessibility holds when the fundamental group is non-virtually solvable.

Center bundle is uniquely integrable if the system is not accessible.

Complete characterization of accessibility classes for systems with 1D center bundle.

Abstract

We consider a partially hyperbolic diffeomorphism $f: M \to M$ without periodic points on a closed manifold $M$. We prove that $f$ is accessible when $M$ is a 3-manifold with non-virtually-solvable fundamental group $\pi_1(M)$. In the case where $\dim E^c = 1$, we demonstrate that the center bundle $E^c$ is uniquely integrable if $f$ lacks accessibility. Furthermore, we provide a complete characterization of accessibility classes for such systems with one-dimensional center bundles.