Partially hyperbolic diffeormorphisms, ergodicity, and transverse foliations in dimension 3

TL;DR

This work provides a complete topological classification of transitive partially hyperbolic diffeomorphisms on closed 3‑manifolds with exponential fundamental group growth, showing they are collapsed Anosov flows (via a semiconjugacy to an Anosov flow). The authors develop a robust framework using two transverse 2‑dimensional foliations by Gromov hyperbolic leaves, proving a dichotomy: either their intersection foliation has a generalized Reeb surface, or it is leafwise quasigeodesic in both foliations; this dichotomy underpins the collapsed Anosov flow structure and the ergodicity results. A key technical achievement is handling orientability and establishing unique integrability of branching cs/cu foliations, which leads to a clean identification of the center foliation and, in the volume-preserving non‑solvable π1 setting, ergodicity (indeed K‑system). The methods yield a broad impact on understanding transverse foliations in 3‑manifolds and provide a concrete route to proving accessibility and ergodicity for a wide class of partially hyperbolic systems.

Abstract

We give a complete topological classification of transitive partially hyperbolic diffeomorphisms in 3-manifolds in terms of Anosov flows, completing a program proposed by Pujals. In particular, this also allows to give a full answer to the ergodicity conjecture of Hertz-Hertz-Ures for partially hyperbolic diffeomorphisms in dimension 3. This is achieved by showing a general result about pairs of transverse $2$-dimensional foliations in 3-manifolds with Gromov hyperbolic leaves which may be of independent interest.

Figures (16)

• Figure 1: In the left a Reeb annulus and its lift to the universal cover of the leaf. In the right, a Reeb crown with its corresponding lift to the universal cover of the leaf. In the orientable case, these are the two instances of generalized Reeb surfaces.
• Figure 2: Non separated rays $r_1,r_2$ in a one dimensional foliation inside a leaf $L \in \widetilde{\mathcal{F}}$. The leaves $\ell_3,\ell_4, \ell_5$ are also non separated from the rays $r_1,r_2$. (See definition of $NS(r_1,r_2)$ in § \ref{['ss.returns']}.)
• Figure 3: When every curve of $\widetilde{\mathcal{G}}$ is quasigeodesic in its corresponding leaf, and the foliation comes from a partially hyperbolic diffeomorphism, one can show that in every leaf one sees a quasigeodesic fan foliation as depicted in the figure (which is also the figure one sees in leaves of topological Anosov flows). Every quasigeodesic line shares one of the endpoints, depicted by $p$.
• Figure 4: The shaded region depicts $U_{r_1,r_2}$.
• Figure 5: If the region $B_{1,n}$ is not trivially foliated, one gets some non separated leaves.

Theorems & Definitions (112)

• Corollary 1.1
• Remark 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Remark 2.5
• Proposition 2.6