Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T13:59:26.964Z Has data issue: false hasContentIssue false
  • English
  • Français

Average co-ordinate entropy

Part of: Measure-theoretic ergodic theory Classical measure theory

Published online by Cambridge University Press:  09 April 2009

Genevieve Mortiss
Genevieve Mortiss
Affiliation:
School of Mathematics, The University of New South Wales, Sydney NSW 2052, Australia e-mail: [email protected]
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.

A notion of entropy for the non-singular action of finite co-ordinate changes on is introduced. This quantity-average co-ordinate or AC entropy-is calculated for product measures and G-measures. It is shown that the type III classes can be subdivided using AC entropy. An equivalence relation is established for which AC entropy is an invariant.

MSC classification

Secondary: 28D20: Entropy and other invariants 28A35: Measures and integrals in product spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 73 , Issue 2 , October 2002 , pp. 171 - 186
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Brown, G. and Dooley, A. H., ‘Odometer actions on G-measures’, Ergodic Theory Dynam. Systems 11 (1991), 279–307.CrossRefGoogle Scholar
[2]Brown, G. and Dooley, A. H., ‘Dichotomy theorems for G-measures’, Internat. J. Math. 5 (1994), 827–834.CrossRefGoogle Scholar
[3]Brown, G., Dooley, A. H. and Lake, J., ‘On the Krieger-Araki-Woods ratio set’, Tohoku Math. J. (2) 47 (1995), 1–13.CrossRefGoogle Scholar
[4]Doob, J. L., Stochastic processes (Wiley, New York, 1953).Google Scholar
[5]Dooley, A. H., Kiemes, I. and Quas, A. N., ‘Product and Markov measures of type III’, J. Aust rat. Math. Soc. Ser. A 65 (1998), 84–110.CrossRefGoogle Scholar
[6]Keane, M., ‘Strongly mixing g-measures’, Invent. Math. 16 (1972), 309–324.CrossRefGoogle Scholar
[7]LePage, R. and Mandrekar, V., ‘On likelihood ratios of measures given by Markov chains’, Proc. Amer. Math. Soc. 52 (1975), 377–380.CrossRefGoogle Scholar
[8]Lodkin, A. A., ‘Absolute continuity of measures corresponding to Markov processes with discrete time’, Theory Probab. Appl. 16 (1971), 690–694.CrossRefGoogle Scholar
[9]Moore, C., ‘Invariant measures on product spaces’, in: Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Vol. II: Contributions to probability theory, Part 2 (Univ. California Press, Berkeley, 1966) pp. 447–459.Google Scholar
[10]Mortiss, G., ‘An invariant for non-singular isomorphism’, Ergodic Theory Dynam. Systems (submitted).Google Scholar
[11]Mortiss, G., Average co-ordinate entropy and a non-singular version of restricted orbit equivalence (Ph.D. Thesis, University of New South Wales, 1997).Google Scholar
[12]Silva, C. E. and Thieullen, P., ‘A skew product entropy for non-singular transformations’, J. London Math. Soc. (2) 52 (1995), 497–516.CrossRefGoogle Scholar