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On the Critical Dimension and AC Entropy for Markov Odometers

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Abstract.

Markov odometers are natural models for non-homogeneous Markov chains, and are natural generalisations of infinite product measures. We show how to calculate the critical dimension of these measures: this is an invariant which describes the asymptotic growth rate of sums of Radon-Nikodym derivatives. This interesting invariant appears to give a kind of entropy for non-singular odometer actions. The techniques require a law of large numbers for inhomogeneous Markov chains.

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References

  • G Brown AH Dooley (1998) ArticleTitleOn G-measures and product measures Ergod Theor Dyn Syst 18 95–107 Occurrence Handle0919.28014 Occurrence Handle1609491 Occurrence Handle10.1017/S0143385798097545

    Article  MATH  MathSciNet  Google Scholar 

  • JL Doob (1953) Stochastic Processes Wiley New York Occurrence Handle0053.26802

    MATH  Google Scholar 

  • Dobrushin RL (1956) The central limit theorem for non-homogeneous Markov chains. Teor Veroyatnost i Primenen 1: 72–89, 365–425

    Google Scholar 

  • AH Dooley (2006) ArticleTitleThe critical dimension: an approach to non-singular entropy Contemporary Math 385 61–76

    Google Scholar 

  • AH Dooley T Hamachi (2003) ArticleTitleNonsingular dynamical systems, Bratteli diagrams and Markov odometers Israel J Math 138 93–123 Occurrence Handle1061.37001 Occurrence Handle2031952

    MATH  MathSciNet  Google Scholar 

  • Dooley AH, Mortiss G On the critical dimension of product odometers. Ergod Theor Dyn Syst, submitted

  • Hartfiel DJ (1998) Markov Set-chains. Lect Notes Math 1695 Berlin Heidelberg New York: Springer

  • D Maharam (1969) ArticleTitleInvariant measures and Radon-Nikodym derivatives Trans Amer Math Soc 135 223–248 Occurrence Handle0172.32804 Occurrence Handle232914 Occurrence Handle10.2307/1995014

    Article  MATH  MathSciNet  Google Scholar 

  • Mortiss G (1997) Average co-ordinate entropy and a non-singular version of restricted orbit equivalence. PhD Thesis, UNSW

  • G Mortiss (2002) ArticleTitleAverage co-ordinate entropy J Austral Math Soc 73 171–186 Occurrence Handle1031.37011 Occurrence Handle1926070 Occurrence Handle10.1017/S144678870000879X

    Article  MATH  MathSciNet  Google Scholar 

  • G Mortiss (2003) ArticleTitleAn invariant for non-singular isomorphism Ergod Theor Dyn Syst 23 885–893 Occurrence Handle1067.37008 Occurrence Handle1992668 Occurrence Handle10.1017/S0143385702001414

    Article  MATH  MathSciNet  Google Scholar 

  • M Rosenblatt-Roth (1964) ArticleTitleSome theorems concerning the strong law of large numbers for non-homogeneous Markov chains Ann Math Statist 35 566–576 Occurrence Handle0127.35402 Occurrence Handle162258

    MATH  MathSciNet  Google Scholar 

  • H Sato (1991) ArticleTitleAbsolute continuity of locally equivalent Markov chains Mem Fac Sci Kyushu Univ Ser A 45 285–308 Occurrence Handle0755.60034 Occurrence Handle1133115

    MATH  MathSciNet  Google Scholar 

  • L Wen (1999) ArticleTitleA limit property of arbitrary discrete information sources Taiwan J Math 3 539–546 Occurrence Handle0954.94005

    MATH  Google Scholar 

  • W Yang (2002) ArticleTitleConvergence in the Cesaro sense and strong law of large numbers for nonhomogeneous Markov chains Linear Algebra Appl 354 275–288 Occurrence Handle1012.60033 Occurrence Handle1927662 Occurrence Handle10.1016/S0024-3795(02)00298-7

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. University of New South Wales, Sydney, Australia

    A. H. Dooley & G. Mortiss

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  1. A. H. Dooley
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  2. G. Mortiss
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Dooley, A., Mortiss, G. On the Critical Dimension and AC Entropy for Markov Odometers. Mh Math 149, 193–213 (2006). https://doi.org/10.1007/s00605-005-0372-6

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  • DOI: https://doi.org/10.1007/s00605-005-0372-6

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