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Non-singular
$\mathbb {Z}^d$
-actions: an ergodic theorem over rectangles with application to the critical dimensions
- Part of
-
- Journal:
- Ergodic Theory and Dynamical Systems / Volume 41 / Issue 12 / December 2021
- Published online by Cambridge University Press:
- 02 December 2020, pp. 3722-3739
- Print publication:
- December 2021
-
- Article
- Export citation
-
We adapt techniques developed by Hochman to prove a non-singular ergodic theorem for
$\mathbb {Z}^d$
-actions where the sums are over rectangles with side lengths increasing at arbitrary rates, and in particular are not necessarily balls of a norm. This result is applied to show that the critical dimensions with respect to sequences of such rectangles are invariants of metric isomorphism. These invariants are calculated for the natural action of
$\mathbb {Z}^d$
on a product of d measure spaces.
