Twisted cohomological equations for translation flows

TL;DR

This paper establishes the existence of solutions to twisted cohomological equations on translation surfaces using harmonic analysis, demonstrating stability of certain product translation flows and their time-{ au} maps.

Contribution

It introduces new methods to solve twisted cohomological equations with controlled derivative loss and proves stability of product translation flows on higher-dimensional translation manifolds.

Findings

Solutions exist with at most 3+ derivatives lost in Sobolev spaces.

Product translation flows on 3D manifolds are stable in Katok's sense.

Time-{ au} maps of translation flows are also stable.

Abstract

We prove by methods of harmonic analysis a result on existence of solutions for twisted cohomological equations on translation surfaces with loss of derivatives at most 3+ in Sobolev spaces. As a consequence we prove that product translation flows on (3-dimensional) translation manifolds which are products of a (higher genus) translation surface with a (flat) circle are stable in the sense of A. Katok. In turn, our result on product flows implies a stability result of time-{\tau} maps of translation flows on translation surfaces.