# Journal of the European Mathematical Society

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**Volume 12, Issue 2, 2010, pp. 365–383**

**DOI: 10.4171/JEMS/201**

A ratio ergodic theorem for multiparameter non-singular actions

Michael Hochman^{[1]}(1) Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Givat Ram, 91904, JERUSALEM, ISRAEL

We prove a ratio ergodic theorem for non-singular free ℤ^{d} and ℝ^{d} actions, along balls in an arbitrary norm. Using a Chacon–Ornstein type lemma the proof is reduced to a statement about the amount of mass of a probability measure that can concentrate on (thickened) boundaries of balls in ℝ^{d}. The proof relies on geometric properties of norms, including the Besicovitch covering lemma and the fact that boundaries of balls have lower dimension than the ambient space. We also show that for general group actions, the Besicovitch covering property not only implies the maximal inequality, but is equivalent to it, implying that further generalization may require new methods.

*Keywords: *Group actions, measure preserving transformations, commuting transformations, nonsingular actions, ergodic theorem, maximal inequality

Hochman Michael: A ratio ergodic theorem for multiparameter non-singular actions. *J. Eur. Math. Soc.* 12 (2010), 365-383. doi: 10.4171/JEMS/201