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twisted cohomological equations

Ergodic Theory and Dynamical Systems (1)

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Twisted cohomological equations for translation flows
Part of
GIOVANNI FORNI
Journal:

Ergodic Theory and Dynamical Systems / Volume 42 / Issue 3 / March 2022

Published online by Cambridge University Press:

22 October 2021, pp. 881-916

Print publication:

March 2022
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We prove by methods of harmonic analysis a result on the 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 (three-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.