Oscillation and the mean ergodic theorem for uniformly convex Banach spaces. (10th January 2014)

• Record Type: Journal Article
• Authors: AVIGAD, JEREMY; RUTE, JASON
• Is Part Of: Ergodic theory and dynamical systems. Volume 35:Number 4(2015:Aug.)
• Page Start: 1009
• Page End: 1027
• Publication Date: 2014-01-10
• Subjects: Ergodic theory -- Periodicals; Differentiable dynamical systems -- Periodicals; 515.42
• DOI: 10.1017/etds.2013.90
• Languages: English
• ISSNs: 0143-3857

Abstract

<abstract abstract-type="normal"> <title>Abstract</title> <p>Let <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjqhzctng" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$ \mathbb{B} $]]></tex-math></alternatives></inline-formula> be a <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjqhzcwqg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$p$]]></tex-math></alternatives></inline-formula>-uniformly convex Banach space, with <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjqhzcx7q" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$p\geq 2$]]></tex-math></alternatives></inline-formula>. Let <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjqhzcx66" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$T$]]></tex-math></alternatives></inline-formula> be a linear operator on <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjqhzcwc9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$ \mathbb{B} $]]></tex-math></alternatives></inline-formula>, and let <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjqhzcxgb" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${A}_{n} x$]]></tex-math></alternatives></inline-formula> denote the ergodic average <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjqhzcx3n" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$(1/ n){\mathop{\sum }\nolimits}_{i\lt n} {T}^{n} x$]]></tex-math></alternatives></inline-formula>. We prove the following variational inequality in the case where <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjqhzcvdb" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$T$]]></tex-math></alternatives></inline-formula> is power bounded from above and below: for any increasing sequence <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjqhzctjx" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathop{({t}_{k} )}\nolimits_{k\in \mathbb{N} } $]]></tex-math></alternatives></inline-formula> of natural numbers we have <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjqhzcw35" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[${\mathop{\sum }\nolimits}_{k} \mathop{\Vert {A}_{{t}_{k+ 1} } x- {A}_{{t}_{k} } x\Vert }\nolimits ^{p} \leq C\mathop{\Vert x\Vert }\nolimits ^{p} $]]></tex-math></alternatives></inline-formula>, where the constant <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjqhzcvb9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$C$]]></tex-math></alternatives></inline-formula> depends only on <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjqhzcvwk" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$p$]]></tex-math></alternatives></inline-formula> and the modulus of uniform convexity. For <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjqhzctcb" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$T$]]></tex-math></alternatives></inline-formula> a non-expansive operator, we obtain a weaker bound on the number of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjqhzcthd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\varepsilon $]]></tex-math></alternatives></inline-formula>-fluctuations in the sequence. We clarify the relationship between bounds on the number of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjqhzcwkd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\varepsilon $]]></tex-math></alternatives></inline-formula>-fluctuations in a sequence and bounds on the rate of metastability, and provide lower bounds on the rate of metastability that show that our main result is sharp.</p> </abstract>