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$\omega $-recurrence in skew products

Published online by Cambridge University Press:  03 April 2013

JON CHAIKA  and
DAVID RALSTON
JON CHAIKA
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, USA email jonchaika@math.uchicago.edu
DAVID RALSTON
Affiliation:
Depatment of Mathematics/CIS, SUNY College at Old Westbury, PO 210, Old Westbury, NY 11568, USA email ralstond@oldwestbury.edu
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Abstract

The rate of recurrence to measurable subsets in a conservative, ergodic infinite-measure-preserving system is quantified by generic divergence or convergence of certain sums given by a function $\omega (n)$. In the context of skew products over transformations of a probability space, we relate this notion to the more frequently studied question of the growth rate of ergodic sums (including Lyapunov exponents). We study in particular skew products over an irrational rotation given by bounded variation $ \mathbb{Z} $-valued functions: first the generic situation is studied and recurrence quantified, and then certain specific skew products over rotations are shown to violate this generic rate of recurrence.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 5 , October 2014 , pp. 1525 - 1537
Copyright
Copyright ©2013 Cambridge University Press 

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