On ergodicity of some cylinder flows

Volume 163 / 2000

Krzysztof Frączek Fundamenta Mathematicae 163 (2000), 117-130 DOI: 10.4064/fm-163-2-117-130

Abstract

We study ergodicity of cylinder flows of the form   $T_f:{\mathbb T}×ℝ → {\mathbb T}×ℝ$, $T_f(x,y) = (x+α,y+f(x))$, where $f:{\mathbb T} → ℝ$ is a measurable cocycle with zero integral. We show a new class of smooth ergodic cocycles. Let k be a natural number and let f be a function such that $D^kf$ is piecewise absolutely continuous (but not continuous) with zero sum of jumps. We show that if the points of discontinuity of $D^kf$ have some good properties, then $T_f$ is ergodic. Moreover, there exists $ε_f > 0$ such that if $v:{\mathbb T}→ℝ$ is a function with zero integral such that $D^kv$ is of bounded variation with $Var(D^kv) < ε_f$, then $T_{f+v}$ is ergodic.

Authors

  • Krzysztof Frączek

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