We exhibit an explicit class of minimal interval exchange maps (i.e.m.’s) T T for which the cohomological equation \[ Ψ − Ψ ∘ T = Φ \Psi -\Psi \circ T=\Phi \] has a bounded solution Ψ \Psi provided that the datum Φ \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 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.

A standard interval exchange map is a one-to-one map of the interval that 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 T0 be a standard interval exchange map of restricted Roth type, and let r be an integer 2. We prove that, amongst C r+3 deformations of T0 that are C r+3 tangent to T0 at the singularities, those that are conjugated to T0 by a C r -diffeomorphism close to the identity form a C 1 -submanifold of codimension (g − 1)(2r + 1) + s.Here, g is the genus and s is the number of marked points of the translation surface obtained by suspension of T0. Both g and s can be computed from the combinatorics of T0.

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ABSTRACT. We prove that the solutions of the cohomological equation for Roth type interval exchange maps are Hölder continuous provided that the datum is of class C r with r > 1 and belongs to a finite-codimension linear subspace.

We consider a one-parameter family of expanding interval maps {Tα} α∈[0,1] (japanese continued fractions) which include the Gauss map (α = 1) and the nearest integer and by-excess continued fraction maps (α = 1 2 , α = 0). We prove that the Kolmogorov-Sinai entropy h(α) of these maps depends continuously on the parameter and that h(α) → 0 as α → 0. Numerical results suggest that this convergence is not monotone and that the entropy function has infinitely many phase transitions and a self-similar structure. Finally, we find the natural extension and the invariant densities of the maps Tα for α = 1 n .