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<p>We consider a special case of the question of classification of invariant Radon measures of ℤ<sup><i>m</i></sup>-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<sup><i>m</i></sup>. 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.</p>

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

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  • 发表日期：

    2025年3月24日

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  • Yuriy Tumarkin

摘要

We consider a special case of the question of classification of invariant Radon measures of ℤm-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 ℝm. 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.