Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T04:49:40.841Z Has data issue: false hasContentIssue false
  • English
  • Français

On the critical dimensions of product odometers

Published online by Cambridge University Press:  01 April 2009

ANTHONY H. DOOLEY  and
GENEVIEVE MORTISS
ANTHONY H. DOOLEY
Affiliation:
School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia (email: [email protected])
GENEVIEVE MORTISS
Affiliation:
School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 2 , April 2009 , pp. 475 - 485
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50). American Mathematical Society, Providence, RI, 1997.CrossRefGoogle Scholar
[2]Aaronson, J. and Weiss, B.. On the asymptotics of a 1-parameter family of infinite measure-preserving transformations. Bol. Soc. Brasil Mat. (NS) 29 (1988), 181–193.CrossRefGoogle Scholar
[3]Doob, J. L.. Stochastic Processes. Wiley, New York, 1953.Google Scholar
[4]Dooley, A. H.. The critical dimension: an approach to non-singular entropy. Contemp. Math. 385 (2004), 65–76.CrossRefGoogle Scholar
[5]Dooley, A. H. and Mortiss, G.. On the critical dimension and AC entropy for Markov odometers. Monatsh. Math. 149 (2006), 193–213.CrossRefGoogle Scholar
[6]Dooley, A. H. and Mortiss, G.. The critical dimensions of Hamachi shifts. Tohôku J. Math. 59 (2007), 57–66.Google Scholar
[7]Fisher, A. M.. Integer Cantor sets and an order-two ergodic theorem. Ergod. Th. & Dynam. Sys. 13 (1992), 45–64.CrossRefGoogle Scholar
[8]Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137–173.CrossRefGoogle Scholar
[9]Maharam, D.. Invariant measures and Radon-Nikodym derivatives. Trans. Amer. Math. Soc. 135 (1969), 223–248.CrossRefGoogle Scholar
[10]Maharam, D.. Incompressible transformations. Fund. Math. 56 (1964), 35–50.CrossRefGoogle Scholar
[11]Mortiss, G.. Average co-ordinate entropy and a non-singular version of restricted orbit equivalence. PhD Thesis, University of New South Wales, 1997.Google Scholar
[12]Mortiss, G.. Average co-ordinate entropy. J. Austral. Math. Soc. 73 (2002), 171–186.CrossRefGoogle Scholar
[13]Mortiss, G.. An invariant for non-singular isomorphism. Ergod. Th. & Dynam. Sys. 23 (2003), 885–893.CrossRefGoogle Scholar
[14]Mortiss, G.. A non-singular inverse Vitali lemma with applications. Ergod. Th. & Dynam. Sys. 20 (2000), 1215–1229.CrossRefGoogle Scholar