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On the asymptotic range of cocyles for shifts of finite type

Published online by Cambridge University Press:  19 September 2008

Zaqueu Coelho
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 20570, São Paulo SP, Brazil

Abstract

We study the problem of lifting ergodicity to certain skew-product extensions over shifts of finite type. This can be done by computing the asymptotic range of the defining cocycle. For a class of ergodic shift-invariant measures and a class of functions (defining conservative extensions) we give a characterization of ergodicity in terms of weights of periodic orbits in the shift space.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1993

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References

[At]Atkinson, G.. Recurrence of cocycles and random walks. J. London Math. Soc. 13 (1976).Google Scholar
[BD]Brown, G. & Dooley, A. H.. Odometer actions on g-measures. Ergod. Th. & Dynam. Sys. 11 (1991), 279307.CrossRefGoogle Scholar
[Bo]Bowen, R.. Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Springer Lecture Notes in Mathematics 470. Springer: Berlin, 1975.CrossRefGoogle Scholar
[Co]Coelho, Z.. PhD Thesis. University of Warwick (1990).Google Scholar
[CP]Coelho, Z. & Parry, W.. Central limit asymptotics for shifts of finite type. Israel J. Maths 69 (1990), 235249.CrossRefGoogle Scholar
[De]Dekking, F. M.. On transience and recurrence of generalised random walks. Z. Wahrsch. verw. Gebiete 61 (1982), 459465.CrossRefGoogle Scholar
[FM]Feldman, J. & Moore, C.. Ergodic equivalence relations, cohomology, and von Neumann algebras I. Trans. Amer. Math. Soc. 234 (1977), 289324.CrossRefGoogle Scholar
[Gu]Guivarc'h, Y.. Propriétés ergodiques, en mesure infinie, de certaines systèmes dynamiques fibrés. Ergod. Th. & Dynam. Sys. 9 (1989), 433453.CrossRefGoogle Scholar
[GH]Guivarc'h, Y. & Hardy, J.. Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov. Ann. Poincaré 24 (1988), 7398.Google Scholar
[KR]Krieger, W.. On the finitary isomorphisms of Markov shifts that have finite expected coding time. Z. Wahrsch. verw. Gebiete 65 (1983), 323328.CrossRefGoogle Scholar
[LE]Ledrappier, F.. Principe variationnel et systèmes dynamiques symboliques. Z. Wahrsch. verw. Gebiete 30 (1974), 185202.CrossRefGoogle Scholar
[Li]Livšic, A. N.. Cohomology of dynamical systems. Math. USSR Izv. 6 (1972), 12781301.CrossRefGoogle Scholar
[MS]Moore, C. & Schmidt, K.. Coboundaries and homomorphisms for nonsingular actions and a problem of H. Helson. Proc. London Math. Soc. 40 (1980), 443475.CrossRefGoogle Scholar
[PS]Parry, W. & Schmidt, K.. Natural coefficients and invariants for Markov shifts. Invent. Math. 76 (1984), 1532.CrossRefGoogle Scholar
[Re]Rees, M.. Checking ergodicity of some geodesic flows with infinite Gibbs measure. Ergod. Th. & Dynam. Sys. 1 (1981), 107133.CrossRefGoogle Scholar
[Ru]Ruelle, D.. Thermodynamic formalism. Addison-Wesley: Reading, MA, 1978.Google Scholar
[Sc1]Schmidt, K.. Cocycles of Ergodic Transformation Groups. Macmillan: India, 1977.Google Scholar
[Sc2]Schmidt, K.. On recurrence. Z. Wahrsch. verw. Gebiete 68 (1984), 7595.CrossRefGoogle Scholar
[Sc3]Schmidt, K.. Hyperbolic structure preserving isomorphisms of Markov shifts. Israel J. Math. 55 (1986), 213228.CrossRefGoogle Scholar
[Sc4]Schmidt, K.. Hyperbolic structure preserving isomorphisms of Markov shifts II. Israel J. Math. 58 (1987), 225242.CrossRefGoogle Scholar
[Sc5]Schmidt, K.. Algebraic ideas in ergodic theory. CBMS Reg. Conf., Amer. Math. Soc. 76 1990.Google Scholar
[Wa1]Walters, P.. Ruelle's operator theorem and g-measures. Trans. Amer. Math. Soc. 214 (1975), 375387.Google Scholar
[Wa2]Walters, P.. An Introduction to Ergodic Theory. Graduate Texts in Maths 79. Springer: New York, 1982.CrossRefGoogle Scholar

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