Hostname: page-component-6898cfcb55-kwdml Total loading time: 0 Render date: 2026-11-11T12:15:33.962Z Has data issue: false hasContentIssue false
  • English
  • Français

Computing the critical dimensions of Bratteli–Vershik systems with multiple edges

Published online by Cambridge University Press:  05 April 2011

ANTHONY H. DOOLEY and
RIKA HAGIHARA
ANTHONY H. DOOLEY
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia (email: a.dooley@unsw.edu.au, r.hagihara@unsw.edu.au)
RIKA HAGIHARA
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia (email: a.dooley@unsw.edu.au, r.hagihara@unsw.edu.au)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

The critical dimension is an invariant that measures the growth rate of the sums of Radon–Nikodym derivatives for non-singular dynamical systems. We show that for Bratteli–Vershik systems with multiple edges, the critical dimension can be computed by a formula analogous to the Shannon–McMillan–Breiman theorem. This extends earlier results of Dooley and Mortiss on computing the critical dimensions for product and Markov odometers on infinite product spaces.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 1 , February 2012, pp. 103 - 117
Copyright
Copyright © Cambridge University Press 2011

References

[1]Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50). American Mathematical Society, Providence, RI, 1998.Google Scholar
[2]Dooley, A. H.. The critical dimension: an approach to non-singular Markov chains. Contemp. Math. 385 (2006), 61–76.Google Scholar
[3]Dooley, A. H. and Hamachi, T.. Nonsingular dynamical systems, Bratteli diagrams and Markov odometers. Israel J. Math. 138 (2003), 93–123.CrossRefGoogle Scholar
[4]Dooley, A. H. and Mortiss, G.. On the critical dimension and AC entropy for Markov odometers. Monatsh. Math. 149(3) (2006), 193–213.Google Scholar
[5]Dooley, A. H. and Mortiss, G.. On the critical dimension of product odometers. Ergod. Th. & Dynam. Sys. 29 (2009), 475–485.Google Scholar
[6]Hartfiel, D. J.. Markov Set-Chains (Lecture Notes in Mathematics, 1695). Springer, Berlin, 1998.Google Scholar
[7]Herman, R. H., Putnam, I. F. and Skau, C. F.. Ordered Bratteli diagrams, dimension groups and topological dynamics. Internat. J. Math. 3(6) (1992), 827–864.Google Scholar
[8]Krieger, W.. On non-singular transformations of a measure space, I, II. Z. Wahrsch. Verw. Gebeite 11 (1969), 83–97; Z. Wahrsch. Verw. Gebeite 11 (1969), 98–119.Google Scholar
[9]Maharam, D.. Invariant measures and Radon–Nikodym derivatives. Trans. Amer. Math. Soc. 135 (1969), 223–248.Google Scholar
[10]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
[11]Mortiss, G.. Average co-ordinate entropy. J. Aust. Math. Soc. 73 (2002), 171–186.Google Scholar
[12]Mortiss, G.. An invariant for non-singular isomorphism. Ergod. Th. & Dynam. Sys. 23 (2003), 885–893.Google Scholar
[13]Ornstein, D.. Ergodic Theory, Randomness, and Dynamical Systems (Yale Mathematical Monographs, 5). Yale University Press, New Haven, CT, 1974.Google Scholar
[14]Rosenblatt-Roth, M.. Some theorems concerning the strong law of large numbers for non-homogeneous Markov chains. Ann. Math. Statist. 35 (1964), 566–576.CrossRefGoogle Scholar
[15]Wen, L.. A limit property of arbitrary discrete information sources. Taiwanese J. Math. 3(4) (1999), 539–546.Google Scholar