Accessibility and central integrability in the absence of periodic points

TL;DR

The paper addresses the accessibility problem for partially hyperbolic diffeomorphisms on closed manifolds with no periodic points, establishing accessibility in dimension three when $ olinebreak[0]$ olinebreak[0] $\,\pi_1(M)$ is not virtually solvable. It develops a thorough center integrability theory for the one‑dimensional center, proving the center bundle is uniquely integrable on the non‑accessible set and that complementary regions form $I$‑bundles, enabling a complete classification of accessibility classes via a codimension‑one lamination $\,\Gamma(f)$. The results yield a clean dichotomy for non‑accessible systems: either $\,\Gamma(f)$ contains infinitely many compact $su$‑leaves or it is (uniquely) minimal, with implications for dynamical coherence and foliation structure; in 3D, this leads to a precise model‑class classification after finite lifts. The findings have implications for the broader classification program of 3‑manifolds, linking accessibility to manifold topology (non‑virtually solvable fundamental groups) and to the geometry of center laminations, including leaf conjugacy to skew products or discretized Anosov flows. Overall, the work advances understanding of how absence of periodic points interacts with center dynamics to shape accessibility and global foliations in partially hyperbolic systems.

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 $π_1(M)$. 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.

Figures (1)

• Figure 1: The $su$-plaques through $\omega_i$ and center curves through $f^{n_i}(y)$ in a coordinate cube

Theorems & Definitions (57)

• Conjecture 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4