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Spatial recurrence for ergodic fractal measures

Volume 248 / 2019

Nadav Dym Studia Mathematica 248 (2019), 1-29 MSC: Primary 28A80. DOI: 10.4064/sm8715-3-2018 Published online: 11 February 2019


We study the invertible version of Furstenberg’s ‘ergodic CP shift systems’, which describe a random walk on measures on Euclidean space. These measures are by definition invariant under a scaling procedure, and satisfy a condition called adaptedness under a ‘local’ translation operation. We show that the distribution is in fact non-singular with respect to a suitably defined translation operator on measures, and derive discrete and continuous pointwise ergodic theorems for the translation action.


  • Nadav DymDepartment of Mathematics
    Weizmann Institute of Science
    Rehovot 7610001, Israel

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