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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: 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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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

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.

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