On uniqueness of G-measures and g-measures

• Autorzy: Ai Hua Fan
• Języki publikacji: EN

Abstrakty

EN

We give a simple proof of the sufficiency of a log-lipschitzian condition for the uniqueness of G-measures and g-measures which were studied by G. Brown, A. H. Dooley and M. Keane. In the opposite direction, we show that the lipschitzian condition together with positivity is not sufficient. In the special case where the defining function depends only upon two coordinates, we find a necessary and sufficient condition. The special case of Riesz products is discussed and the Hausdorff dimension of Riesz products is calculated.

Słowa kluczowe

EN

G-measures; g-measures; ergodic measures; Riesz products; quasi-invariance; dimension of measures

Kategorie tematyczne

• 43A05: Measures on groups and semigroups, etc.
• 28D05: Measure-preserving transformations
• 28A35: Measures and integrals in product spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1996
• Tom: 119
• Numer: 3
• Strony: 255-269

Daty

wydano

1996

otrzymano

1995-11-22

poprawiono

1996-03-13

Twórcy

autor

Ai Hua Fan

• Department of Mathematics, University of Cergy-Pontoise, 2, Avenue A. Chauvin, 95302 Cergy-Pontoise, France

Bibliografia

• [1] L. Breiman, Probability, Addison-Wesley, 1968.
• [2] G. Brown and A. H. Dooley, Odometer actions on G-measures, Ergodic Theory Dynam. Systems 11 (1991), 279-307.
• [3] Y. S. Chow and H. Teicher, Probability Theory, 2nd ed., Springer Texts Statist., Springer, 1988.
• [4] A. H. Fan, Sur les dimensions de mesures, Studia Math. 111 (1994), 1-17.
• [5] C. C. Graham and O. C. McGehee, Essays in Commutative Harmonic Analysis, Springer, 1979.
• [6] B. Host, J. F. Méla et F. Parreau, Analyse harmonique des mesures, Astérisque 135-136 (1986).
• [7] B. Jamison, Asymptotic behavior of successive iterates of continuous functions under a Markov operator, J. Math. Anal. Appl. 9 (1964), 203-214.
• [8] M. Keane, Strongly mixing g-measures, Invent. Math. 16 (1974), 309-324.
• [9] F. Ledrappier, Principe variationnel et systèmes dynamiques symboliques, Z. Wahrsch. Verw. Gebiete 30 (1974), 185-202.
• [10] K. Petersen, Ergodic Theory, Cambridge Univ. Press, 1983.
• [11] B. Petit, g-mesures et schémas de Bernoulli, Thèse de troisième cycle, Université de Rennes, 1974.
• [12] J. Peyrière, Etude de quelques propriétés des produits de Riesz, Ann. Inst. Fourier (Grenoble) 25 (2) (1975), 127-169.
• [13] J. Peyrière, Mesures singulières associées à des découpages aléatoires d'un hypercube, Colloq. Math. 51 (1987), 267-276.
• [14] W. Rudin, Functional Analysis, McGraw-Hill, 1991.
• [15] E. Seneta, Non-negative Matrices and Markov Chains, Springer Ser. Statist., Springer, 1981.
• [16] P. Walters, Ruelle's operator theorem and g-measures, Trans. Amer. Math. Soc. 214 (1975), 375-387.