78 research outputs found
Uniformly Hyperbolic Finite-Valued SL(2,R)-Cocycles
We consider finite families of SL(2,R) matrices whose products display
uniform exponential growth. These form open subsets of (SL(2,R))^N, and we
study their components, boundary, and complement. We also consider the more
general situation where the allowed products of matrices satisfy a Markovian
rule.Comment: 64 pages, 16 figure
Linearization of generalized interval exchange maps
A standard interval exchange map is a one-to-one map of the interval which is
locally a translation except at finitely many singularities. We define for such
maps, in terms of the Rauzy-Veech continuous fraction algorithm, a diophantine
arithmetical condition called restricted Roth type which is almost surely
satisfied in parameter space. Let be a standard interval exchange map of
restricted Roth type, and let be an integer . We prove that,
amongst deformations of which are tangent to at
the singularities, those which are conjugated to by a
diffeomorphism close to the identity form a submanifold of codimension
. Here, is the genus and is the number of marked points
of the translation surface obtained by suspension of . Both and
can be computed from the combinatorics of .Comment: 52 pages. This version includes a new section where we explain how to
adapt our result to the setting of perturbations of linear flows on
translation surface
On the cohomological equation for interval exchange maps
We exhibit an explicit full measure class of minimal interval exchange maps T
for which the cohomological equation has a bounded
solution provided that the datum belongs to a finite codimension
subspace of the space of functions having on each interval a derivative of
bounded variation.
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.
Contents
0. French abridged version
1. Interval exchange maps and the cohomological equation. Main Theorem
2. Rauzy--Veech--Zorich continued fraction algorithm and its acceleration
3. Special Birkhoff sums
4. The Diophantine condition
5. Sketch of the proof of the theoremComment: 11 pages, french abstract and abridged versio
Exponential mixing for the Teichmuller flow
We study the dynamics of the Teichmuller flow in the moduli space of Abelian
differentials (and more generally, its restriction to any connected component
of a stratum). We show that the (Masur-Veech) absolutely continuous invariant
probability measure is exponentially mixing for the class of Holder
observables. A geometric consequence is that the \SL(2,\R) action in the
moduli space has a spectral gap.Comment: 49 page
The cohomological equation for Roth type interval exchange maps
We exhibit an explicit full measure class of minimal interval exchange maps T
for which the cohomological equation has a bounded
solution provided that the datum belongs to a finite codimension
subspace of the space of functions having on each interval a derivative of
bounded variation. 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.
CONTENTS 0. Introduction 1. The continued fraction algorithm for interval
exchange maps 1.1 Interval exchnge maps 1.2 The continued fraction algorithm
1.3 Roth type interval exchange maps 2. The cohomological equation 2.1 The
theorem of Gottschalk and Hedlund 2.2 Special Birkhoff sums 2.3 Estimates for
functions of bounded variation 2.4 Primitives of functions of bounded variation
3. Suspensions of interval exchange maps 3.1 Suspension data 3.2 Construction
of a Riemann surface 3.3 Compactification of 3.4 The cohomological
equation for higher smoothness 4. Proof of full measure for Roth type 4.1 The
basic operation of the algorithm for suspensions 4.2 The Teichm\"uller flow 4.3
The absolutely continuous invariant measure 4.4 Integrability of 4.5 Conditions (b) and (c) have full measure 4.6 The main step
4.7 Condition (a) has full measure 4.8 Proof of the Proposition Appendix A
Roth--type conditions in a concrete family of i.e.m. Appendix B A non--uniquely
ergodic i.e.m. satsfying condition (a) ReferencesComment: 64 pages, 4 figures (jpeg files
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