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Sobolev regularity of solutions of the cohomological equation

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

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 gforni@math.umd.edu
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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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References

Athreya, J.. Quantitative recurrence and large deviations for Teichmuller geodesic flow. Geom. Dedicata 119 (2006), 121–140.CrossRefGoogle Scholar
Avila, A. and Viana, M.. Simplicity of Lyapunov spectra: proof of the Zorich–Kontsevich conjecture. Acta Math. 198(1) (2007), 1–56.CrossRefGoogle Scholar
Bers, L.. Spaces of degenerating Riemann surfaces. Discontinuous Groups and Riemann Surfaces, Proc. Conf. Univ. of Maryland, College Park, Md, 1973 (Princeton, NJ) (Annals of Mathematical Studies, 79) . Princeton University Press, Princeton, NJ, 1974, pp. 43–55.Google Scholar
Barreira, L. and Pesin, Y.. Smooth ergodic theory and non-uniformly hyperbolic dynamics. Handbook of Dynamical Systems, 1B. Eds. Hasselblatt, B. and Katok, A.. Elsevier, Amsterdam, 2005.Google Scholar
Chaika, J. and Eskin, A.. Every flat surface is Birkhoff and Oseledets generic in almost every direction. J. Mod. Dyn. 9 (2015), 1–23.CrossRefGoogle Scholar
Chen, D. and Möller, M.. Quadratic differentials in low genus: exceptional and non-varying strata. Ann. Sci. Éc. Norm Supér (4) 47(2) (2014), 309–369.CrossRefGoogle Scholar
Collet, P., Epstein, H. and Gallavotti, G.. Perturbations of geodesic flows on surfaces of constant negative curvature and their mixing properties. Comm. Math. Phys. 95 (1984), 61–112.CrossRefGoogle Scholar
de la Llave, R., Marco, J. M. and Moriyón, R.. Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation. Ann. of Math. (2) 123 (1986), 537–611.CrossRefGoogle Scholar
Dolgopyat, D.. Livsic theory for compact group extensions of hyperbolic systems. Mosc. Math. J. 5(1) (2005), 55–67.CrossRefGoogle Scholar
Eskin, A. and Mirzakhani, M.. Invariant and stationary measures for the SL(2, ℝ) action on moduli space. Publ. Math. Inst. Hautes Études Sci. 127(1) (2018), 95–324.CrossRefGoogle Scholar
Farkas, H. M. and Kra, I.. Riemann Surfaces, 2nd edn. Springer, New York, 1992.CrossRefGoogle Scholar
Filip, S.. Semisimplicity and rigidity of the Kontsevich–Zorich cocycle. Invent. Math. 205(3) (2016), 617–670.CrossRefGoogle Scholar
Filip, S.. Notes on the multiplicative ergodic theorem. Ergod. Th. & Dynam. Sys. 39(5) (2019), 1153–1189.CrossRefGoogle Scholar
Flaminio, L., Forni, G. and Rodriguez Hertz, F.. Invariant distributions for homogeneous flows and affine transformations. J. Mod. Dyn. 10 (2016), 33–79.CrossRefGoogle Scholar
Flaminio, L. and Forni, G.. Invariant distributions and time averages for horocycle flows. Duke Math. J. 119(3) (2003), 465–526.Google Scholar
Flaminio, L. and Forni, G.. Equidistribution of nilflows and applications to theta sums. Ergod. Th. & Dynam. Sys. 26(02) (2006), 409–433.CrossRefGoogle Scholar
Flaminio, L. and Forni, G.. On the cohomological equation for nilflows. J. Mod. Dyn. 1(1) (2007), 37–60.CrossRefGoogle Scholar
Forni, G.. Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus. Ann. of Math. (2) 146(2) (1997), 295–344.CrossRefGoogle Scholar
Forni, G.. Deviation of ergodic averages for area-preserving flows on surfaces of higher genus. Ann. of Math. (2) 155(1) (2002), 1–103.CrossRefGoogle Scholar
Forni, G.. On the Lyapunov exponents of the Kontsevich–Zorich Cocycle. Handbook of Dynamical Systems, 1B. Eds. Hasselblatt, B. and Katok, A.. Elsevier, Amsterdam, 2005.Google Scholar
Forni, G.. Sobolev regularity of solutions of the cohomological equation. Preprint, 2007, arXiv:0707.0940v2.Google Scholar
Forni, G.. A geometric criterion for the nonuniform hyperbolicity of the Kontsevich–Zorich cocycle. J. Mod. Dyn. 5(2) (2011), 355–395; with an appendix by Carlos Matheus.CrossRefGoogle Scholar
Forni, G. and Matheus, C.. Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards. J. Mod. Dyn. 8(3–4) (2014), 271–436.CrossRefGoogle Scholar
Gottschalk, W. H. and Hedlund, G. A.. Topological Dynamics (AMS Colloquium Publications, 36) . American Mathematical Society, Providence, RI, 1955.CrossRefGoogle Scholar
Guillemin, V. and Kazhdan. Some inverse spectral results for negatively curved 2-manifolds. Topology 19 (1980), 301–312.CrossRefGoogle Scholar
Gutiérrez-Romo, R.. Simplicity of the Lyapunov spectra of certain quadratic differentials. Preprint, 2017, arXiv:1711.02006.Google Scholar
Herman, M. R.. Sur les courbes invariantes par les difféomorphismes de l’anneau, Vol. 1 (Astérisque) . Société Mathématique de France, Paris, France, 1983, pp. 103–104.Google Scholar
Herman, M. R.. Simple proofs of local conjugacy theorems for diffeomorphisms of the circle with almost every rotation number. Bol. Soc. Bras. Mat. 16(1) (1985), 45–83.CrossRefGoogle Scholar
Hubert, P. and Schmidt, T.. Affine diffeomorphisms and the Veech dichotomy. Handbook of Dynamical Systems, 1B. Eds. Hasselblatt, B. and Katok, A.. Elsevier, Amsterdam, 2005.Google Scholar
Katok, A. and Kononenko, A.. Cocycles’ stability for partially hyperbolic systems. Math. Res. Lett. 3(2) (1996), 191–210.CrossRefGoogle Scholar
Kerckhoff, S. P., Masur, H. and Smillie, J.. Ergodicity of billiard flows and quadratic differentials. Ann. of Math. (2) 124 (1986), 293–311.CrossRefGoogle Scholar
Kontsevich, M.. Lyapunov exponents and Hodge theory. The Mathematical Beauty of Physics: A memorial volume for Claude Itzykson (Advanced Series in Mathematical Physics, 24) . World Scientific, Singapore, 1997, pp. 318–332.Google Scholar
Kontsevich, M. and Zorich, A.. Connected components of the moduli space of abelian differentials with prescribed singularities. Inv. Math. 153 (2003), 631–678.CrossRefGoogle Scholar
Lanneau, E.. Connected components of the moduli spaces of quadratic differentials. Ann. Sci. Éc. Norm. Supér. (4) 41(1) (2008), 1–56.CrossRefGoogle Scholar
Livsic, A.. Homology properties of U-systems. Math. Notes USSR Acad. Sci. 10 (1971), 758–763 (in Russian).Google Scholar
Lions, J. L. and Magenes, E.. Problèmes aux limites non homogènes et applications, Vol. 1. Dunod, Paris, 1968.Google Scholar
Masur, H.. Interval exchange transformations and measured foliations. Ann. of Math. (2) 115 (1982), 168–200.CrossRefGoogle Scholar
Masur, H.. Logarithmic law for geodesics in moduli space. Mapping Class Groups and Moduli Spaces of Riemann Surfaces, Proceedings, Göttingen June–August 1991 (Providence, RI) (Contemporary Mathematics, 150) . American Mathematical Society, Providence, RI, 1993, pp. 229–245.Google Scholar
Masur, H.. Ergodic theory of translation surfaces. Handbook of Dynamical Systems, 1B. Eds. Hasselblatt, B. and Katok, A.. Elsevier, Amsterdam, 2005.Google Scholar
Marmi, S., Moussa, P. and Yoccoz, J.-C.. On the cohomological equation for interval exchange maps. C. R. Math. Acad. Sci. Paris 336 (2003), 941–948.CrossRefGoogle Scholar
Marmi, S., Moussa, P. and Yoccoz, J.-C.. The cohomological equation for Roth type interval exchange maps. J. Amer. Math. Soc. 18 (2005), 823–872.CrossRefGoogle Scholar
Marmi, S., Moussa, P. and Yoccoz, J.-C.. Linearization of generalized interval exchange maps. Ann. of Math. (2) 176(03) (2012), 1583–1646.CrossRefGoogle Scholar
Marmi, S. and Yoccoz, J.-C.. Hölder regularity of the solutions of the cohomological equation for Roth type interval exchange maps. Comm. Math. Phys. 344(1) (2016), 117–139.CrossRefGoogle Scholar
Masur, H. and Tabachnikov, S.. Rational Billiards and Flat Structures. Handbook of Dynamical Systems, 1A. Eds. Hasselblatt, B. and Katok, A.. Elsevier, Amsterdam, 2005, pp. 1015–1089.Google Scholar
Nag, S.. The Complex Analytic Theory of Teichmüller Spaces. John Wiley and Sons, New York, 1988.Google Scholar
Nelson, E.. Analytic vectors. Ann. of Math. (2) 70 (1959), 572–615.CrossRefGoogle Scholar
Treviño, R.. On the non-uniform hyperbolicity of the Kontsevich–Zorich cocycle for quadratic differentials. Geom. Dedicata 163(1) (2013), 311–338.CrossRefGoogle Scholar
Veech, W.. Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. (2) 115 (1982), 201–242.CrossRefGoogle Scholar
Veech, W.. The Teichmüller geodesic flow. Ann. of Math. (2) 124 (1986), 441–530.CrossRefGoogle Scholar
Veech, W.. Periodic points and invariant pseudomeasures for toral endomorphisms. Ergod. Th. & Dynam. Sys. 6(3) (1986), 449–473.CrossRefGoogle Scholar
Veech, W.. Moduli spaces of quadratic differentials. J. Anal. Math. 55 (1990), 117–171.CrossRefGoogle Scholar
Wilkinson, A.. The Cohomological Equation for Partially Hyperbolic Diffeomorphisms (Astérisque, 358) . Société Mathématique de France, Paris, France, 2013, pp. 75–165.Google Scholar
Zorich, A.. Asymptotic flag of an orientable measured foliation on a surface. Geometric Study of Foliations: Proceedings of the International Symposium/Workshop. Eds. Mizutani, T., Masuda, K., Matsumoto, S., Inaba, T., Tsuboi, T. and Mitsumatsu, Y.. World Scientific, River Edge, NJ, 1994, pp. 479–498.Google Scholar
Zorich, A.. Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents. Ann. Inst. Fourier (Grenoble) 46 (1996), 325–370.CrossRefGoogle Scholar
Zorich, A.. Deviation for interval exchange transformations. Ergod. Th. & Dynam. Sys. 17 (1997), 1477–1499.CrossRefGoogle Scholar
Zorich, A.. How do the leaves of a closed 1-form wind around a surface? Pseudoperiodic Topology (American Mathematical Society Transl. (2), 197) . Eds. Kontsevich, M., Arnol’d, V. I. and Zorich, A.. American Mathematical Society, Providence, RI, 1999, pp. 135–178.Google Scholar