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The critical dimension for $G$-measures

Published online by Cambridge University Press:  21 October 2015

DANIEL F. MANSFIELD  and
ANTHONY H. DOOLEY
DANIEL F. MANSFIELD
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, Australia email daniel.mansfield@unsw.edu.au
ANTHONY H. DOOLEY
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath, UK email a.h.dooley@bath.ac.uk
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Abstract

The critical dimension of an ergodic non-singular dynamical system is the asymptotic growth rate of sums of consecutive Radon–Nikodým derivatives. This has been shown to equal the average coordinate entropy for product odometers when the size of individual factors is bounded. We extend this result to $G$-measures with an asymptotic bound on the size of individual factors. Furthermore, unlike von Neumann–Krieger type, the critical dimension is an invariant property on the class of ergodic $G$-measures.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 3 , May 2017 , pp. 824 - 836
Copyright
© Cambridge University Press, 2015 

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