On the cohomological equation for interval exchange maps

TL;DR

This paper identifies a large class of minimal interval exchange maps satisfying a Diophantine condition where the cohomological equation admits bounded solutions for a broad class of functions, extending understanding of dynamical systems.

Contribution

It provides an explicit full measure class of interval exchange maps with bounded solutions to the cohomological equation under a Diophantine condition, linking dynamics and number theory.

Findings

Existence of bounded solutions for the cohomological equation in a full measure class.

Characterization of interval exchange maps via a Roth-type Diophantine condition.

Application of Rauzy--Veech--Zorich continued fraction algorithm to analyze solutions.

Abstract

We exhibit an explicit full measure class of minimal interval exchange maps 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 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 theorem