# On ergodicity of some cylinder flows

Fundamenta Mathematicae (2000)

- Volume: 163, Issue: 2, page 117-130
- ISSN: 0016-2736

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topFrączek, Krzysztof. "On ergodicity of some cylinder flows." Fundamenta Mathematicae 163.2 (2000): 117-130. <http://eudml.org/doc/212433>.

@article{Frączek2000,

abstract = {We study ergodicity of cylinder flows of the form
$T_f:\{T\}×ℝ → \{T\}×ℝ$, $T_f(x,y) = (x+α,y+f(x))$,
where $f:\{T\} → ℝ$ 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 $v:\{T\}→ℝ$ 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.},

author = {Frączek, Krzysztof},

journal = {Fundamenta Mathematicae},

keywords = {cylinder flow; ergodicity; piecewise absolutely continuous cocycle},

language = {eng},

number = {2},

pages = {117-130},

title = {On ergodicity of some cylinder flows},

url = {http://eudml.org/doc/212433},

volume = {163},

year = {2000},

}

TY - JOUR

AU - Frączek, Krzysztof

TI - On ergodicity of some cylinder flows

JO - Fundamenta Mathematicae

PY - 2000

VL - 163

IS - 2

SP - 117

EP - 130

AB - We study ergodicity of cylinder flows of the form
$T_f:{T}×ℝ → {T}×ℝ$, $T_f(x,y) = (x+α,y+f(x))$,
where $f:{T} → ℝ$ 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 $v:{T}→ℝ$ 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.

LA - eng

KW - cylinder flow; ergodicity; piecewise absolutely continuous cocycle

UR - http://eudml.org/doc/212433

ER -

## References

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- [6] M. Lemańczyk, F. Parreau and D. Volný, Ergodic properties of real cocycles and pseudo-homogeneous Banach spaces, Trans. Amer. Math. Soc. 348 (1996), 4919-4938. Zbl0876.28021
- [7] W. Parry, Topics in Ergodic Theory, Cambridge Univ. Press, Cambridge, 1981. Zbl0449.28016
- [8] D. Pask, Skew products over the irrational rotation, Israel J. Math. 69 (1990), 65-74. Zbl0703.28009
- [9] D. Pask, Ergodicity of certain cylinder flows, ibid. 76 (1991), 129-152. Zbl0764.28011
- [10] K. Schmidt, Cocycles of Ergodic Transformation Groups, Macmillan Lectures in Math. 1, Delhi, 1977. Zbl0421.28017

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