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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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The cohomological equation for Roth-type interval exchange maps
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by S. Marmi, P. Moussa and J.-C. Yoccoz PDF
J. Amer. Math. Soc. 18 (2005), 823-872 Request permission

Abstract:

We exhibit an explicit class of minimal interval exchange maps (i.e.m.’s) $T$ for which the cohomological equation \[ \Psi -\Psi \circ T=\Phi \] 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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Additional Information
  • S. Marmi
  • Affiliation: Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
  • MR Author ID: 120140
  • 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
  • Received by editor(s): April 7, 2004
  • Published electronically: May 25, 2005
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 18 (2005), 823-872
  • MSC (2000): Primary 37A20; Secondary 11K50, 32G15, 37A45, 37E05
  • DOI: https://doi.org/10.1090/S0894-0347-05-00490-X
  • MathSciNet review: 2163864