UNIQUENESS, ERGODICITY AND UNIDIMENSIONALITY OF INVARIANT MEASURES UNDER A MARKOV OPERATOR

doi:10.1016/S0252-9602(09)60105-3

Abstract

Abstract:

Let X be a compact metric space and C(X) be the space of all continuous functions on X. In this article, the authors
consider the Markov operator T:C(X)^N → C(X)^N defined by
Tf = (∑_j=1^Np_1jf_j ο w_1j, …, ∑_j=1^Np_Njf_jο w_Nj)
for any f =(f ₁, f ₂, …, f _N), where (p _ij) is a N × N transition probability matrix and {w _ij} is an family of continuous transformations on X. The authors study the uniqueness, ergodicity and unidimensionality of T*-invariant measures where T* is the adjoint operator of T.

Key words: Markov operator, invariant measure, ergodicity, unidimensionality

CLC Number:

37A30

TANG Jun-Min, ZHANG Ji-Hong, ZHANG Xiong-Ying. UNIQUENESS, ERGODICITY AND UNIDIMENSIONALITY OF INVARIANT MEASURES UNDER A MARKOV OPERATOR[J].Acta mathematica scientia,Series B, 2009, 29(5): 1309-1322.

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References

[1] Breiman L. The strong law of large numbers for a class of Markov chains. Ann Math Statist, 1960, 31: 801--803

[2] Barnsley M F, Demko S G. Iterated function systems and the global constructions of fractals. Proc R Soc London A, 1985, 399: 243--275

[3] Barnsley M F, Demko S G, Elton J H, Geromino J S. Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilies. Ann Inst H Poincar\'{e}-Prob Stat, 1988, 24(3): 367--394

4] Falconer K J. Fractal Geometry. New York: Wiley, 1990

[5] Fan Aihua. Sur les dimensions de mesures. Studia Math, 1994, 111: 1--17

[6] Fan Aihua. Ergodicity,unidimensionality and multifractality of self-similar measures. Kyushu J Math, 1996, 50: 541--574

[7] Fan Aihua, Lau K S. Iterated function system and Ruelle operator. J Math Anal Appl, 1999, 231: 319--344

[8] Huchinson J E. Fractals and self-similarity. Indiana Univ Math J, 1981, 30: 713--747

[9] Lau K S, Ngai S M. $L^q$-spectrum of the Bernoulli convolution associated with the golden ratio. Studia Math, 1998, 131: 225--251

[10] Lau K S, Ngai S M, Rao Hui. Iterated function systems with overlaps and self-simalar measures. J London Math Soc, 2001, 63: 99--116

[11] Leung C W. Vector-valued submarkov operators and recurrent iterated function systems. J Math Anal Appl, 2001, 254: 12--22

[12] Jamison B. Asymptotic behavior of successive iterates of continuous functions under a Markov operator. J Math Anal Appl, 1964, 9: 203--214

[13] Loève M. Probability Theory II. Berlin: Springer-Verlag, 1978

[14] Peres Y, Solomyak B. Self-similar measures and intersections of Cantor sets. Trans Amer Math Soc, 1998, 350: 4065--4087

[15] Revuz D. Markov Chains. Amsterdam: North-Holland, 1984

[16] Rudin W. Functional Analysis. 2nd ed. New York: Wiley, 1991

[17] Strichartz R S. Self-similar measures and their Fourier transforms III. Indiana Univ Math J, 1993, 42: 367--411

[18] Stichartz R S, Taylor A, Zhang T. Densities of self-similar measures on the line. Experiment Math, 1995, 4: 101--128

[19] Tamashiro M. Dimensions in a seperable metric space. Kyushu J Math, 1995, 49: 143--162

[20] Ye Yuanling. Vector-valued Ruelle operators. J Math Anal Appl, 2004, 299: 341--356

[21] Yu Jinghu, Fan Wentao. Fractal properties of statistically self-similar sets. Acta Mathematica Scientia, 2000, 20B(2): 256--260

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