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Conservativity of random Markov fibred systems

Published online by Cambridge University Press:  01 February 2008

MANFRED DENKER ,
YURI KIFER  and
MANUEL STADLBAUER
MANFRED DENKER
Affiliation:
Institut für Mathematische Stochastik, Maschmühlenweg 8-10, 37073 Göttingen, Germany (email: denker@math.uni-goettingen.de, stadelba@math.uni-goettingen.de)
YURI KIFER
Affiliation:
Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, 91904, Israel (email: kifer@math.huji.ac.il)
MANUEL STADLBAUER
Affiliation:
Institut für Mathematische Stochastik, Maschmühlenweg 8-10, 37073 Göttingen, Germany (email: denker@math.uni-goettingen.de, stadelba@math.uni-goettingen.de)
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Abstract

In this paper we extend results concerning conservativity and the existence of σ-finite measures to random transformations which admit a countable relative Markov partition. We consider random systems which are locally fibre-preserving and which admit a countable, relative Markov partition. If the system is relative irreducible and satisfies a relative distortion property we deduce that the system is either totally dissipative or conservative and ergodic. For conservative systems, we provide sufficient conditions for the existence of absolutely continuous σ-finite invariant measures.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 1 , February 2008 , pp. 67 - 85
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50). American Mathematical Society, Providence, RI, 1997.CrossRefGoogle Scholar
[2]Aaronson, J., Denker, M. and Urbański, M.. Ergodic theory for Markov fibred systems and parabolic rational maps. Trans. Amer. Math. Soc. 337(2) (1993), 495–548.CrossRefGoogle Scholar
[3]Adler, R. and Flatto, L.. Geodesic flows, interval maps and symbolic dynamics. Bull. Amer. Math. Soc., (N.S.) 25(2) (1991), 229–334.CrossRefGoogle Scholar
[4]Arnold, L.. Random Dynamical Systems (Springer Monographs in Mathematics). Springer, Berlin, 1998.CrossRefGoogle Scholar
[5]Beardon, A. F.. The Geometry of Discrete Groups (Graduate Texts in Mathematics, 91). Springer, New York, 1983.CrossRefGoogle Scholar
[6]Beardon, A. F.. Iteration of Rational Functions (Graduate Texts in Mathematics, 132). Springer, New York, 1991.CrossRefGoogle Scholar
[7]Bowen, R. and Series, C.. Markov maps associated with Fuchsian groups. Publ. Math. Inst. Hautes Études. Sci. 50 (1979), 153–170.CrossRefGoogle Scholar
[8]Cogburn, R.. The ergodic theory of Markov chains in random environments. Z. Wahrsch. Verw. Gebiete 66 (1984), 109–128.CrossRefGoogle Scholar
[9]Denker, M. and Gordin, M.. Gibbs measures for fibered systems. Adv. Math. 48(2) (1999), 161–192.CrossRefGoogle Scholar
[10]Denker, M. and Holzmann, H.. Markov partitions for fibre expanding systems. Colloq. Math. (2006) Preprint, to appear.Google Scholar
[11]Epstein, D. and Petronio, C.. An exposition of Poincare’s polyhedron theorem. Enseign. Math., II. Ser. 40 (1994), 113–170.Google Scholar
[12]Kac, M.. On the notion of recurrence in discrete stochastic processes. Bull. Amer. Math. Soc. 53 (1947), 1002–1010.CrossRefGoogle Scholar
[13]Kifer, Y.. Ergodic Theory of Random Transformations (Progress in Probability and Statistics, 10). Birkhäuser, Boston, MA, 1986.CrossRefGoogle Scholar
[14]Kifer, Y.. Perron–Frobenius theorem, large deviations, and random perturbations in random environments. Math. Z. 222 (1996), 677–698.CrossRefGoogle Scholar
[15]Orey, S.. Markov chains with stochastically stationary transition probabilities. Ann. Probab. 19 (1991), 907–928.CrossRefGoogle Scholar
[16]Series, C.. Geometrical Markov coding of geodesics on surfaces of constant negative curvature. Ergod. Th. & Dynam. Sys. 6 (1986), 601–625.CrossRefGoogle Scholar
[17]Stadlbauer, M.. The return sequence of the Bowen–Series map associated to punctured surfaces. Fund. Math. 182 (2004), 221–240.CrossRefGoogle Scholar
[18]Stadlbauer, M. and Stratmann, B. O.. Infinite ergodic theory for Kleinian groups. Ergod. Th. & Dynam. Sys. 25 (2005), 1305–1323.CrossRefGoogle Scholar