The cohomological equation for Roth-type interval exchange maps

Abstract

We exhibit an explicit class of minimal interval exchange maps (i.e.m.’s) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which the cohomological equation <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Psi minus normal upper Psi ring upper T equals normal upper Phi"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Ψ</mml:mi> <mml:mo>−</mml:mo> <mml:mi mathvariant="normal">Ψ</mml:mi> <mml:mo>∘</mml:mo> <mml:mi>T</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">Φ</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\Psi -\Psi \circ T=\Phi</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> has a bounded solution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Psi"> <mml:semantics> <mml:mi mathvariant="normal">Ψ</mml:mi> <mml:annotation encoding="application/x-tex">\Psi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> provided that the datum <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi"> <mml:semantics> <mml:mi mathvariant="normal">Φ</mml:mi> <mml:annotation encoding="application/x-tex">\Phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. 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.