Metric convergence and metric isomorphism

Abstract 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.

Download Full-text

On the critical dimensions of product odometers

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385708000606 ◽

2009 ◽

Vol 29 (2) ◽

pp. 475-485 ◽

Cited By ~ 3

Author(s):

ANTHONY H. DOOLEY ◽

GENEVIEVE MORTISS

Keyword(s):

Critical Dimension ◽

Order Of Growth ◽

Critical Dimensions ◽

Covering Lemma ◽

Singular Action ◽

Metric Isomorphism

AbstractMortiss 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.

Download Full-text