Ergodic measures for periodic type ℤm-skew-products over interval exchange transformations

Abstract We consider a special case of the question of classification of invariant Radon measures of ℤ m -valued skew-products over interval exchange transformations, which arise as Poincaré 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 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.