Ergodic measures for periodic type $\mathbb{Z}^m$-skew-products over Interval Exchange Transformations

Abstract

We consider a special case of the question of classification of invariant Radon measures of $\mathbb{Z}^m$-valued skew-products over interval exchange transformations, which arise as Poincar\'e sections of the linear flow on periodic infinite translation surfaces. In the case of periodic type skew-products, we obtain a full classification of ergodic invariant Radon measures, showing them to be precisely the Maharam measures, a family of measures parametrised by $\mathbb{R}^m$. For the proof we translate Rauzy-Veech renormalisation for skew-products into the symbolic language of the adic coding, and apply a symbolic result of Aaronson, Nakada, Sarig and Solomyak. Further, we use this language and a new extension of the Rauzy-Veech cocycle to find an explicit form for the Maharam measures and deduce the weak*-continuity of the measures depending on the parameter.