On ergodicity of some cylinder flows

Frączek, Krzysztof

Fundamenta Mathematicae, Tome 163 (2000), p. 117-130 / Harvested from The Polish Digital Mathematics Library

Résumé

We study ergodicity of cylinder flows of the form $T_{f} : T \times ℝ \to T \times ℝ$ , $T_{f} (x, y) = (x + α, y + f (x))$ , 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 $ε_{f} > 0$ such that if $v : T \to ℝ$ is a function with zero integral such that $D^{k} v$ is of bounded variation with $V a r (D^{k} v) < ε_{f}$ , then $T_{f + v}$ is ergodic.

Publié le : 2000-01-01

Zbl 0979.37005

EUDML-ID : urn:eudml:doc:212433

@article{bwmeta1.element.bwnjournal-article-fmv163i2p117bwm,
     author = {Krzysztof Fr\k aczek},
     title = {On ergodicity of some cylinder flows},
     journal = {Fundamenta Mathematicae},
     volume = {163},
     year = {2000},
     pages = {117-130},
     zbl = {0979.37005},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv163i2p117bwm}
}

Frączek, Krzysztof. On ergodicity of some cylinder flows. Fundamenta Mathematicae, Tome 163 (2000) pp. 117-130. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv163i2p117bwm/

Bibliographie

[00000] [1] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer, Berlin, 1982.

[00001] [2] H. Furstenberg, Strict ergodicity and transformations on the torus, Amer. J. Math. 83 (1961), 573-601. | Zbl 0178.38404

[00002] [3] P. Gabriel, M. Lemańczyk et P. Liardet, Ensemble d'invariants pour les produits croisés de Anzai, Mém. Soc. Math. France 47 (1991). | Zbl 0754.28011

[00003] [4] P. Hellekalek and G. Larcher, On the ergodicity of a class of skew products, Israel J. Math. 54 (1986), 301-306. | Zbl 0609.28007

[00004] [5] M. R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Mat. IHES 49 (1979), 5-234.

[00005] [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. | Zbl 0876.28021

[00006] [7] W. Parry, Topics in Ergodic Theory, Cambridge Univ. Press, Cambridge, 1981. | Zbl 0449.28016

[00007] [8] D. Pask, Skew products over the irrational rotation, Israel J. Math. 69 (1990), 65-74. | Zbl 0703.28009

[00008] [9] D. Pask, Ergodicity of certain cylinder flows, ibid. 76 (1991), 129-152. | Zbl 0764.28011

[00009] [10] K. Schmidt, Cocycles of Ergodic Transformation Groups, Macmillan Lectures in Math. 1, Delhi, 1977. | Zbl 0421.28017