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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)


The cohomological equation for Roth-type interval exchange maps

Authors: S. Marmi, P. Moussa and J.-C. Yoccoz
Journal: J. Amer. Math. Soc. 18 (2005), 823-872
MSC (2000): Primary 37A20; Secondary 11K50, 32G15, 37A45, 37E05
Published electronically: May 25, 2005
MathSciNet review: 2163864
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Abstract | References | Similar Articles | Additional Information

Abstract: We exhibit an explicit class of minimal interval exchange maps (i.e.m.'s) $T$ for which the cohomological equation

\begin{displaymath}\Psi-\Psi\circ T=\Phi \end{displaymath}

has a bounded solution $\Psi$ provided that the datum $\Phi $ belongs to a finite codimension subspace of the space of functions having on each interval a derivative of bounded variation. The proof is purely dynamical and is based on a renormalization argument and on Gottshalk-Hedlund's theorem. If the datum is more regular the loss of differentiability in solving the cohomological equation will be the same.

The class of interval exchange maps is characterized in terms of a diophantine condition of Roth type imposed to an acceleration of the Rauzy-Veech-Zorich continued fraction expansion associated to $T$. More precisely one must impose a growth rate condition for the matrices appearing in the continued fraction algorithm together with a spectral gap condition (which guarantees unique ergodicity) and a coherence condition. We also prove that the set of Roth-type interval exchange maps has full measure.

In the appendices we construct concrete examples of Roth-type i.e.m.'s and we show how the growth rate condition alone does not imply unique ergodicity.

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  • [Ar] P. Arnoux ``Ergodicité générique des billiards polygonaux [d'après Kerckhoff, Masur, Smillie]'' Séminaire Bourbaki n. 696, Astérisque 161-162 (1988) 203-221. MR 0992210 (90c:58097)
  • [Bo] M. Boshernitzan ``A condition for minimal interval exchange maps to be uniquely ergodic'' Duke Math. J. 52 (1985) 723-752.MR 0808101 (87i:28012)
  • [Ch] Y. Cheung ``Hausdorff dimension of the set of nonergodic directions. With an appendix by M. Boshernitzan'' Ann. of Math. 158 (2003) 661-678. MR 2018932 (2004k:37069)
  • [Co] J. Coffey ``Some remarks concerning an example of a minimal, non-uniquely ergodic interval exchange transformation'' Math. Z. 199 (1988) 577-580.MR 0968323 (90c:28025)
  • [FLP] A. Fathi, F. Laudenbach and V. Poenaru ``Travaux de Thurston sur les surfaces'' Astérisque 66-67 (1979).MR 0568308 (82m:57003)
  • [Fo1] G. Forni ``Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus'' Annals of Mathematics 146 (1997) 295-344. MR 1477760 (99d:58102)
  • [Fo2] G. Forni ``Deviation of ergodic averages for area-preserving flows on surfaces of higher genus'' Annals of Mathematics 155 (2002) 1-103.MR 1888794 (2003g:37009)
  • [Fo3] G. Forni, private communication (2003).
  • [GH] W.H. Gottschalk and G.A. Hedlund ``Topological dynamics'' American Mathematical Society Colloquium Publications, 36. American Mathematical Society, Providence, RI (1955). MR 0074810 (17:650e)
  • [KH] A. Katok and B. Hasselblatt ``Introduction to the modern theory of dynamical systems'' Encyclopedia of Mathematics and its Applications 54, Cambridge University Press, (1995). MR 1326374 (96c:58055)
  • [KS] A. Katok and A.M. Stepin ``Approximations in Ergodic Theory'' Russ. Math. Surv. 22 (1967) 77-102. MR 0219697 (36:2776)
  • [Ke1] M. Keane ``Interval exchange transformations'' Math. Z. 141 (1975) 25-31.MR 0357739 (50:10207)
  • [Ke2] M. Keane ``Non-ergodic interval exchange transformations'' Isr. J. Math. 26 (1977) 188-196.MR 0435353 (55:8313)
  • [Ker] S. P. Kerckhoff ``Simplicial systems for interval exchange maps and measured foliations'' Ergod. Th. Dynam. Sys. 5 (1985) 257-271.MR 0796753 (87g:58075)
  • [KMS] S. Kerckhoff, H. Masur and J. Smillie ``Ergodicity of billiard flows and quadratic differentials'' Ann. of Math. 124 (1986) 293-311.MR 0855297 (88f:58122)
  • [KN] H. B. Keynes and D. Newton ``A ``Minimal'', Non-uniquely Ergodic Interval Exchange Transformation'' Math. Z. 148 (1976) 101-105.MR 0409766 (53:13518)
  • [KR] M. Keane and G. Rauzy ``Stricte ergodicité des échanges d'intervalles'' Math. Z. 174 (1980) 203-212.MR 0593819 (82d:28014)
  • [KZ] M. Kontsevich and A. Zorich ``Connected components of the moduli spaces of Abelian differentials with prescribed singularities'' Inv. Math. 153 (2003) 631-678. MR 2000471 (2005b:32030)
  • [Ma] H. Masur ``Interval exchange transformations and measured foliations'' Annals of Mathematics 115 (1982) 169-200. MR 0644018 (83e:28012)
  • [MMY] S. Marmi, P. Moussa and J.-C. Yoccoz ``On the cohomological equation for interval exchange maps'' C. R. Math. Acad. Sci. Paris 336 (2003) 941-948.MR 1994599 (2004i:37003)
  • [MS] H. Masur and J. Smillie ``Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms'' Comment. Math. Helv. 68 (1993) 289-307.MR 1214233 (94d:32028)
  • [Ra] G. Rauzy ``Échanges d'intervalles et transformations induites'' Acta Arith. (1979) 315-328.MR 0543205 (82m:10076)
  • [Re] M. Rees ``An alternative approach to the ergodic theory of measured foliations'' Ergod. Th. Dyn. Sys. 1 (1981) 461-488.MR 0662738 (84c:58065)
  • [Ta] S. Tabachnikov ``Billiards'' Panoramas et Synthèses, S.M.F. 1 (1995).MR 1328336 (96c:58134)
  • [V1] W. Veech ``Interval exchange transformations'' Journal d'Analyse Mathématique 33 (1978) 222-272. MR 0516048 (80e:28034)
  • [V2] W. Veech ``Gauss measures for transformations on the space of interval exchange maps'' Ann. of Math. 115 (1982) 201-242. MR 0644019 (83g:28036b)
  • [V3] W. Veech ``The metric theory of interval exchange transformations I. Generic spectral properties'' Amer. J. of Math. 106 (1984) 1331-1359.MR 0765582 (87j:28024a)
  • [V4] W. Veech ``The metric theory of interval exchange transformations II. Approximation by primitive interval exchanges'' Amer. J. of Math. 106 (1984) 1361-1387.MR 0765583 (87j:28024b)
  • [V5] W. Veech ``The metric theory of interval exchange transformations III. The Sah Arnoux Fathi invariant'' Amer. J. of Math. 106 (1984) 1389-1421.MR 0765584 (87j:28024c)
  • [V6] W. Veech ``The Teichmüller geodesic flow'' Ann. of Math. 124 (1986) 441-530.MR 0866707 (88g:58153)
  • [V7] W. Veech ``Moduli spaces of quadratic differentials'' Journal d'Analyse Mathématique 55 (1990) 117-171.MR 1094714 (92e:32014)
  • [Y] J.-C. Yoccoz ``Continued fraction algorithms for interval exchange maps: an introduction'' preprint (2004), to appear in the proceedings of conference Frontiers in Number Theory, Physics and Geometry, Les Houches, 9 - 21 March 2003.
  • [Z1] A. Zorich ``Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents'' Annales de l'Institut Fourier Tome 46 fasc. 2 (1996) 325-370.MR 1393518 (97f:58081)
  • [Z2] A. Zorich ``Deviation for interval exchange transformations'' Ergod. Th. Dyn. Sys. 17 (1997), 1477-1499.MR 1488330 (99e:58124)
  • [Z3] A. Zorich ``On Hyperplane Sections of Periodic Surfaces'' Amer. Math. Soc. Translations 179 (1997), 173-189. MR 1437163 (98h:58148)
  • [Z4] A. Zorich ``How Do the Leaves of a Closed $1$-form Wind Around a Surface?'' in Pseudoperiodic Topology, V. Arnold, M. Kontsevich and A. Zorich editors, Amer. Math. Soc. Translations 197 (1999) 135-178.MR 1733872 (2001c:57019)

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Additional Information

S. Marmi
Affiliation: Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy

P. Moussa
Affiliation: Service de Physique Théorique, CEA/Saclay, 91191 Gif-Sur-Yvette, France

J.-C. Yoccoz
Affiliation: Collège de France, 3, Rue d’Ulm, 75005 Paris, France

PII: S 0894-0347(05)00490-X
Received by editor(s): April 7, 2004
Published electronically: May 25, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.