On the critical dimensions of product odometers

Reference:

Dooley, A. H. and Mortiss, G., 2009. On the critical dimensions of product odometers. Ergodic Theory and Dynamical Systems, 29 (02), pp. 475-485.

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Official URL:

http://dx.doi.org/10.1017/S0143385708000606

Abstract

Mortiss introduced the notion of critical dimension of a non-singular action, a measure of the order of growth of sums of Radon derivatives. The critical dimension was shown to be an invariant of metric isomorphism; this invariant was calculated for two-point product odometers and shown to coincide, in certain cases, with the average coordinate entropy. In this paper we extend the theory to apply to all product odometers, introduce upper and lower critical dimensions, and prove a Katok-type covering lemma.

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