2004/03/30 by Stefano Marmi, Marmi, Stefano, Pierre Moussa +3 · 1 citation
Mathematics · Physics and Astronomy · #Chaos control and synchronization #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Quantum chaos and dynamical systems #math.CV #math.DS #math.NT
paper · pdf · doi:10.48550/arxiv.math/0403518
64 pages, 4 figures (jpeg files)
arxiv created 2004/03/30 · openalex publication_date 2004/03/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We exhibit an explicit full measure class of minimal interval exchange maps T for which the cohomological equation Ψ-Ψ∘ T=Φ has a bounded solution Ψ provided that the datum Φ 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 Mζ^* 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 Teichmüller flow 4.3 The absolutely continuous invariant measure 4.4 Integrability of log\Vert Z(1)\Vert 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) References