Smooth generalized interval exchange transformations with wandering intervals, from explicit derived from pseudo-Anosov maps

Abstract

Abstract Starting from any pseudo-Anosov map ϕ on a surface of genus g ⩾ 2 , we construct explicitly a family of Derived from pseudo-Anosov maps f by adapting the construction of Smale’s derived from Anosov maps on the two-torus. This is done by perturbing ϕ at some fixed points. We first consider perturbations at every conical fixed point and then at regular fixed points. We establish the existence of a measure µ, supported by the non-trivial unique minimal component of the stable foliation of f, with respect to which f is mixing. In the process, we construct a uniquely ergodic generalized interval exchange transformation (GIET) with a wandering interval that is semi-conjugated to a self-similar interval exchange transformation. This GIET is obtained as the Poincaré map of a flow renormalized by f which parametrizes stable foliation. When f is , the flow and the GIET are .