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Sobolev regularity of solutions of the cohomological equation

Part of: Ergodic theory Smooth dynamical systems: general theory Low-dimensional dynamical systems

Published online by Cambridge University Press:  10 January 2020

GIOVANNI FORNI
GIOVANNI FORNI*
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD20742-4015, USA email [email protected]
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Abstract

In this survey we prove the sharpest results on the loss of Sobolev regularity for solutions of the cohomological equation for translation flows on translation surfaces, available to the methods developed by the author in Forni [Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus. Ann. of Math. (2)146(2) (1997), 295–344] and Forni [Deviation of ergodic averages for area-preserving flows on surfaces of higher genus. Ann. of Math. (2)155(1) (2002), 1–103]. The paper was mostly written between 2005 and 2006 while the author was at the University of Toronto, Canada, and was posted on arXiv in July 2007 [Forni. Sobolev regularity of solutions of the cohomological equation. Preprint, 2007, arXiv:0707.0940v2]. In an updated introduction we describe our results, taking into account later work on the problem and relevant recent progress in the field of Teichmüller dynamics, interval exchange transformations and translation flows.

Keywords

invariant distributionscohomological equationTeichmüller flowKontsevich–Zorich cocycledistributional cocycles

MSC classification

Primary: 37E35: Flows on surfaces 37C10: Vector fields, flows, ordinary differential equations
Secondary: 37A20: Orbit equivalence, cocycles, ergodic equivalence relations 37C15: Topological and differentiable equivalence, conjugacy, invariants, moduli, classification
Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 3 , March 2021 , pp. 685 - 789
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Copyright
© Cambridge University Press, 2020

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