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A non-singular version of the Oseledeč ergodic theorem

Part of: Ergodic theory

Published online by Cambridge University Press:  24 February 2022

ANTHONY H. DOOLEY Open the ORCID record for ANTHONY H. DOOLEY [Opens in a new window]  and
JIE JIN
ANTHONY H. DOOLEY*
Affiliation:
School of Mathematical and Physical Sciences, University of Technology Sydney, Ultimo 2007, Australia (e-mail: Jie.Jin-3@student.uts.edu.au)
JIE JIN
Affiliation:
School of Mathematical and Physical Sciences, University of Technology Sydney, Ultimo 2007, Australia (e-mail: Jie.Jin-3@student.uts.edu.au)
*
e-mail: anthony.dooley@uts.edu.au
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Abstract

Kingman’s subadditive ergodic theorem is traditionally proved in the setting of a measure-preserving invertible transformation T of a measure space $(X, \mu )$ . We use a theorem of Silva and Thieullen to extend the theorem to the setting of a not necessarily invertible transformation, which is non-singular under the assumption that $\mu $ and $\mu \circ T$ have the same null sets. Using this, we are able to produce versions of the Furstenberg–Kesten theorem and the Oseledeč ergodic theorem for products of random matrices without the assumption that the transformation is either invertible or measure-preserving.

Keywords

non-singular ergodic theoryrandom ergodic theoremcritical dimension

MSC classification

Primary: 37A40: Nonsingular (and infinite-measure preserving) transformations 37A30: Ergodic theorems, spectral theory, Markov operators
Secondary: 37A50: Relations with probability theory and stochastic processes
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 3 , March 2023 , pp. 873 - 886
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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