On uniqueness of G-measures and g-measures

Ai Fan

Studia Mathematica (1996)

  • Volume: 119, Issue: 3, page 255-269
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Fan, Ai. "On uniqueness of G-measures and g-measures." Studia Mathematica 119.3 (1996): 255-269. <http://eudml.org/doc/216299>.

@article{Fan1996,
abstract = {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.},
author = {Fan, Ai},
journal = {Studia Mathematica},
keywords = {G-measures; g-measures; ergodic measures; Riesz products; quasi-invariance; dimension of measures; -measures; log-Lipschitzian condition; uniqueness; measures},
language = {eng},
number = {3},
pages = {255-269},
title = {On uniqueness of G-measures and g-measures},
url = {http://eudml.org/doc/216299},
volume = {119},
year = {1996},
}

TY - JOUR
AU - Fan, Ai
TI - On uniqueness of G-measures and g-measures
JO - Studia Mathematica
PY - 1996
VL - 119
IS - 3
SP - 255
EP - 269
AB - 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.
LA - eng
KW - G-measures; g-measures; ergodic measures; Riesz products; quasi-invariance; dimension of measures; -measures; log-Lipschitzian condition; uniqueness; measures
UR - http://eudml.org/doc/216299
ER -

References

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  1. [1] L. Breiman, Probability, Addison-Wesley, 1968. 
  2. [2] G. Brown and A. H. Dooley, Odometer actions on G-measures, Ergodic Theory Dynam. Systems 11 (1991), 279-307. Zbl0739.58032
  3. [3] Y. S. Chow and H. Teicher, Probability Theory, 2nd ed., Springer Texts Statist., Springer, 1988. 
  4. [4] A. H. Fan, Sur les dimensions de mesures, Studia Math. 111 (1994), 1-17. 
  5. [5] C. C. Graham and O. C. McGehee, Essays in Commutative Harmonic Analysis, Springer, 1979. Zbl0439.43001
  6. [6] B. Host, J. F. Méla et F. Parreau, Analyse harmonique des mesures, Astérisque 135-136 (1986). Zbl0589.43001
  7. [7] B. Jamison, Asymptotic behavior of successive iterates of continuous functions under a Markov operator, J. Math. Anal. Appl. 9 (1964), 203-214. Zbl0133.10701
  8. [8] M. Keane, Strongly mixing g-measures, Invent. Math. 16 (1974), 309-324. 
  9. [9] F. Ledrappier, Principe variationnel et systèmes dynamiques symboliques, Z. Wahrsch. Verw. Gebiete 30 (1974), 185-202. Zbl0276.93004
  10. [10] K. Petersen, Ergodic Theory, Cambridge Univ. Press, 1983. 
  11. [11] B. Petit, g-mesures et schémas de Bernoulli, Thèse de troisième cycle, Université de Rennes, 1974. 
  12. [12] J. Peyrière, Etude de quelques propriétés des produits de Riesz, Ann. Inst. Fourier (Grenoble) 25 (2) (1975), 127-169. Zbl0302.43003
  13. [13] J. Peyrière, Mesures singulières associées à des découpages aléatoires d'un hypercube, Colloq. Math. 51 (1987), 267-276. Zbl0639.60018
  14. [14] W. Rudin, Functional Analysis, McGraw-Hill, 1991. 
  15. [15] E. Seneta, Non-negative Matrices and Markov Chains, Springer Ser. Statist., Springer, 1981. Zbl0471.60001
  16. [16] P. Walters, Ruelle's operator theorem and g-measures, Trans. Amer. Math. Soc. 214 (1975), 375-387. Zbl0331.28013

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