Hostname: page-component-6bf8c574d5-t27h7 Total loading time: 0 Render date: 2025-02-17T13:01:34.801Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected], [email protected])
YURI KIFER
Affiliation:
Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, 91904, Israel (email: [email protected])
MANUEL STADLBAUER
Affiliation:
Institut für Mathematische Stochastik, Maschmühlenweg 8-10, 37073 Göttingen, Germany (email: [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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