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On uniform convergence in ergodic theorems for a class of skew product transformations
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Brettschneider, Julia (2011) On uniform convergence in ergodic theorems for a class of skew product transformations. Discrete and Continuous Dynamical Systems, Vol.29 (No.3). pp. 873-891. doi:10.3934/dcds.2011.29.873 ISSN 1078-0947.
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Official URL: http://dx.doi.org/10.3934/dcds.2011.29.873
Abstract
Consider a class of skew product transformations consisting of an ergodic or a periodic transformation on a probability space (M, B, mu) in the base and a semigroup of transformations on another probability space (Omega, F, P) in the fibre. Under suitable mixing conditions for the fibre transformation, we show that the properties ergodicity, weakly mixing, and strongly mixing are passed on from the base transformation to the skew product (with respect to the product measure). We derive ergodic theorems with respect to the skew product on the product space.
The main aim of this paper is to establish uniform convergence with respect to the base variable for the series of ergodic averages of a function F on M x Omega along the orbits of such a skew product. Assuming a certain growth condition for the coupling function, a strong mixing condition on the fibre transformation, and continuity and integrability conditions for F, we prove uniform convergence in the base and L-p (P)-convergence in the fibre. Under an equicontinuity assumption on F we further show P-almost sure convergence in the fibre. Our work has an application in information theory: It implies convergence of the averages of functions on random fields restricted to parts of stair climbing patterns de fined by a direction.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||
Library of Congress Subject Headings (LCSH): | Ergodic theory, Convergence, Dynamics | ||||
Journal or Publication Title: | Discrete and Continuous Dynamical Systems | ||||
Publisher: | American Institute of Mathematical Sciences | ||||
ISSN: | 1078-0947 | ||||
Official Date: | 10 March 2011 | ||||
Dates: |
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Volume: | Vol.29 | ||||
Number: | No.3 | ||||
Number of Pages: | 19 | ||||
Page Range: | pp. 873-891 | ||||
DOI: | 10.3934/dcds.2011.29.873 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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