Geodesic
A ratio ergodic theorem for multiparameter non-singular actions
Journal of the European Mathematical Society, Tome 12 (2010) no. 2, pp. 365-383 Cet article a éte moissonné depuis la source EMS Press

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We prove a ratio ergodic theorem for non-singular free Zd and Rd 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 Rd. 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.
DOI : 10.4171/jems/201
Classification : 22-XX, 37-XX, 47-XX, 00-XX
Keywords: Group actions, measure preserving transformations, commuting transformations, nonsingular actions, ergodic theorem, maximal inequality 
@article{JEMS_2010_12_2_a3,
     author = {Michael Hochman},
     title = {A ratio ergodic theorem for multiparameter non-singular actions},
     journal = {Journal of the European Mathematical Society},
     pages = {365--383},
     year = {2010},
     volume = {12},
     number = {2},
     doi = {10.4171/jems/201},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/201/}
}
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Michael Hochman. A ratio ergodic theorem for multiparameter non-singular actions. Journal of the European Mathematical Society, Tome 12 (2010) no. 2, pp. 365-383. doi: 10.4171/jems/201

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