The Cohomological Equation for Jointly Integrable Partially Hyperbolic Diffeomorphisms on 3-Manifolds

TL;DR

This work extends Livšic-type results to jointly integrable partially hyperbolic diffeomorphisms on 3-manifolds with virtually solvable fundamental groups by identifying the periodic cycle functional (PCF) as the key obstruction to solving the cohomological equation $\varphi = u\circ f - u + c$. It proves that, for DA- and AB-systems (with a Diophantine center rotation in the AB case), a continuous solution exists if and only if $\varphi$ has trivial PCF, and it establishes Hölder regularity of the solution and higher $C^r$-regularity along central leaves under suitable $k$-bunching conditions; the center direction is where regularity may be lost. The DA-systems are analyzed via conjugacy to linear automorphisms and cocycle techniques, while AB-systems require Diophantine control to handle small denominators, yielding a precise regularity scale. These results provide a unified criterion for solvability and regularity of the cohomological equation in this non-accessible setting and contribute to rigidity questions for low-dimensional dynamical systems with solvable fundamental groups.

Abstract

For a jointly integrable partially hyperbolic diffeomorphism $f$ on a 3-manifold $M$ with virtually solvable fundamental group which satisfies Diophantine condition along the center foliation, we show that the cohomological equation $\varphi = u\circ f - u + c$ has a continuous solution $u$ if and only if $\varphi$ has trivial periodic cycle functional.

Theorems & Definitions (33)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• Theorem 1.8
• Remark 1.9
• Corollary 1.10