The twisted cohomological equation over the geodesic flow

TL;DR

This paper investigates the twisted cohomological equation over geodesic flow on a specific homogeneous space, providing characterizations of obstructions, smooth solutions, and tame Sobolev estimates, advancing the understanding of rigidity in dynamical systems.

Contribution

It characterizes obstructions, constructs smooth solutions, and establishes tame Sobolev estimates for the twisted cohomological equation over geodesic flow, aiding future rigidity applications.

Findings

Characterization of obstructions to solving the equation

Construction of smooth solutions with tame estimates

Tame splittings for non-homogeneous equations

Abstract

We study the twisted cohomoligical equation over the geodesic flow on $SL(2,\mathbb{R})/\Gamma$. We characterize the obstructions to solving the twisted cohomological equation, construct smooth solution and obtain the tame Sobolev estimates for the solution, i.e, there is finite loss of regularity (with respect to Sobolev norms) between the twisted coboundary and the solution. We also give a tame splittings for non-homogeneous cohomological equations. The result can be viewed as a first step toward the application of KAM method in obtaining differential rigidity for partially hyperbolic actions in products of rank-one groups in future works.