$\omega $-recurrence in skew products. (3rd April 2013)

• Record Type: Journal Article
• Authors: CHAIKA, JON; RALSTON, DAVID
• Is Part Of: Ergodic theory and dynamical systems. Volume 34:Number 5(2014:Oct.)
• Page Start: 1525
• Page End: 1537
• Publication Date: 2013-04-03
• Subjects: Ergodic theory -- Periodicals; Differentiable dynamical systems -- Periodicals; 515.42
• DOI: 10.1017/etds.2013.15
• Languages: English
• ISSNs: 0143-3857

Abstract

<abstract abstract-type="normal"> <title>Abstract</title> <p>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 <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1916c78h" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\omega (n)$]]></tex-math></alternatives></inline-formula>. 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 <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1916c89k" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$ \mathbb{Z} $]]></tex-math></alternatives></inline-formula>-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.</p> </abstract>