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.
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,General Mathematics
Reference56 articles.
1. Invariant and stationary measures for the SL(2, ℝ) action on moduli space;Eskin;Publ. Math. Inst. Hautes Études Sci.,2018
2. [GR17] Gutiérrez-Romo, R. . Simplicity of the Lyapunov spectra of certain quadratic differentials. Preprint, 2017, arXiv:1711.02006.
3. Gauss measures for transformations on the space of interval exchange maps;Veech;Ann. of Math. (2),1982
4. Notes on the multiplicative ergodic theorem;Filip;Ergod. Th. and Dynam. Sys.,2019
Cited by
7 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献
1. New phenomena in deviation of Birkhoff integrals for locally Hamiltonian flows;Journal für die reine und angewandte Mathematik (Crelles Journal);2024-01-02
2. Defects in homogenization theory;Séminaire Laurent Schwartz — EDP et applications;2023-12-04
3. A priori bounds for GIETs, affine shadows and rigidity of foliations in genus two;Publications mathématiques de l'IHÉS;2023-10-31
4. Ergodic and Spectral Theory of Area-Preserving Flows on Surfaces;Encyclopedia of Complexity and Systems Science Series;2023
5. Twisted translation flows and effective weak mixing;Journal of the European Mathematical Society;2022-07-12