Accessibility and Ergodicity of Partially Hyperbolic Diffeomorphisms without Periodic Points

TL;DR

This work settles the Strong Ergodicity Conjecture for $C^2$ conservative partially hyperbolic diffeomorphisms on closed 3-manifolds with no periodic points by proving ergodicity through a detailed accessibility analysis. The authors develop a comprehensive geometric framework built on invariant foliations, their completeness, and their intersections, and they distinguish uniform versus non-uniform regimes via a robust theory around a universal circle and ideal boundary dynamics. A central innovation is showing that non-accessibility leads to strong topological constraints (torus bundles or suspension Anosov-type dynamics) incompatible with the assumed fundamental-group conditions, thereby enforcing accessibility and, consequently, ergodicity. The results substantially advance the Hertz–Hertz–Ures ergodicity program by eliminating the aperiodic case under broad topological hypotheses and illuminating how foliation geometry enforces statistical behavior in partially hyperbolic dynamics.

Abstract

We prove that every $C^2$ conservative partially hyperbolic diffeomorphism of a closed 3-manifold without periodic points is ergodic, which gives an affirmative answer to the Ergodicity Conjecture by Hertz-Hertz-Ures in the absence of periodic points. We also show that a partially hyperbolic diffeomorphism of a closed 3-manifold $M$ with no periodic points is accessible if the non-wandering set is all of $M$ and the fundamental group $π_1(M)$ is not virtually solvable.

Figures (14)

• Figure 1: Deck transformations $g_i$ map sequences in $L$ converging to $\xi$ onto $L_0 \cup \partial_\infty L_0$.
• Figure 2: Angles and wedges under deck transformations. Complementary wedges collapse to a single ideal point $a$.
• Figure 3: Configurations of unstable leaves
• Figure 4: Stable ray $l^s_+(x)$ intersects only one of $\gamma_0^+$ or $\gamma_0^-$ on the cylinder
• Figure 5: A discontinous curve

Theorems & Definitions (102)

• Conjecture 1.1: Weak Ergodicity Conjecture
• Conjecture 1.2: Strong Ergodicity Conjecture
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Proposition 2.4
• Proposition 2.5
• Theorem 2.6