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Twisted cohomological equations for translation flows

Part of: Ergodic theory Two-dimensional theory Smooth dynamical systems: general theory Miscellaneous topics of analysis in the complex domain Low-dimensional dynamical systems

Published online by Cambridge University Press:  22 October 2021

GIOVANNI FORNI
GIOVANNI FORNI*
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD, USA
*
e-mail: gforni@umd.edu
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Abstract

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.

Keywords

translation surfaces and flowstwisted cohomological equationstwisted invariant distributionsSobolev estimates

MSC classification

Primary: 37C10: Vector fields, flows, ordinary differential equations 37E35: Flows on surfaces 37A20: Orbit equivalence, cocycles, ergodic equivalence relations
Secondary: 30E20: Integration, integrals of Cauchy type, integral representations of analytic functions 31A20: Boundary behavior (theorems of Fatou type, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 3: Anatole Katok Memorial Issue Part 2: Special Issue of Ergodic Theory and Dynamical Systems , March 2022 , pp. 881 - 916
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

Dedicated to Anatole Katok, who taught us how to think

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