On ergodicity of some cylinder flows

Fundamenta Mathematicae (2000)

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

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Abstract

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We study ergodicity of cylinder flows of the form   ${T}_{f}:T×ℝ\to T×ℝ$, ${T}_{f}\left(x,y\right)=\left(x+\alpha ,y+f\left(x\right)\right)$, where $f:T\to ℝ$ 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 ${D}^{k}f$ is piecewise absolutely continuous (but not continuous) with zero sum of jumps. We show that if the points of discontinuity of ${D}^{k}f$ have some good properties, then ${T}_{f}$ is ergodic. Moreover, there exists ${\epsilon }_{f}>0$ such that if $v:T\to ℝ$ is a function with zero integral such that ${D}^{k}v$ is of bounded variation with $Var\left({D}^{k}v\right)<{\epsilon }_{f}$, then ${T}_{f+v}$ is ergodic.

How to cite

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Frączek, 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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1. [1] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer, Berlin, 1982.
2. [2] H. Furstenberg, Strict ergodicity and transformations on the torus, Amer. J. Math. 83 (1961), 573-601. Zbl0178.38404
3. [3] P. Gabriel, M. Lemańczyk et P. Liardet, Ensemble d'invariants pour les produits croisés de Anzai, Mém. Soc. Math. France 47 (1991). Zbl0754.28011
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5. [5] M. R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Mat. IHES 49 (1979), 5-234.
6. [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. [7] W. Parry, Topics in Ergodic Theory, Cambridge Univ. Press, Cambridge, 1981. Zbl0449.28016
8. [8] D. Pask, Skew products over the irrational rotation, Israel J. Math. 69 (1990), 65-74. Zbl0703.28009
9. [9] D. Pask, Ergodicity of certain cylinder flows, ibid. 76 (1991), 129-152. Zbl0764.28011
10. [10] K. Schmidt, Cocycles of Ergodic Transformation Groups, Macmillan Lectures in Math. 1, Delhi, 1977. Zbl0421.28017

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