A dynamical view of Roth type numbers

A classical theorem by Roth asserts that all algebraic numbers satisfy a Diophantine inequality with exponent 2 + epsilon for any positive epsilon. The set of numbers with this approximation property is called Roth type: it has full measure and it is invariant under the natural GL(2,Z) action. It is less known that Roth type numbers can also be equivalently characterized by means of the regularity of the solutions of the cohomological equation associated to an irrational rotation on the circle.

We will consider a generalization of Roth type numbers to the context of interval exchange transformations (which generalize rotations). We prove that the solutions of the cohomological equation for Roth type interval exchange maps are H\"older continuous provided that the datum is of class $C^r$, with $r>1$, and belongs to a finite-codimension linear subspace. The talk is based on a joint work with J-C Yoccoz and on previous results obtained also in collaboration with P. Moussa.