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 Mosc. Math. J., 2006, Volume 6, Number 4, Pages 657–672 (Mi mmj264)

On the ergodicity of cylindrical transformations given by the logarithm

a Université Paris 13
b Nikolaus Copernicus University

Abstract: Given $\alpha\in[0,1]$ and $\varphi\colon\mathbb T\to\mathbb R$ measurable, the cylindrical cascade $S_{\alpha\varphi}$ is the map from $\mathbb T\times\mathbb R$ to itself given by $S_{\alpha\varphi}(x,y)=(x+\alpha, y+\varphi(x))$, which naturally appears in the study of some ordinary differential equations on $\mathbb R^3$. In this paper, we prove that for a set of full Lebesgue measure of $\alpha\in[0,1]$ the cylindrical cascades $S_{\alpha\varphi}$ are ergodic for every smooth function $\varphi$ with a logarithmic singularity, provided that the average of $\varphi$ vanishes.
Closely related to $S_{\alpha\varphi}$ are the special flows constructed above $R_\alpha$ and under $\varphi+c$, where $c\in\mathbb R$ is such that $\varphi+c>0$. In the case of a function $\varphi$ with an asymmetric logarithmic singularity, our result gives the first examples of ergodic cascades $S_{\alpha\varphi}$ with the corresponding special flows being mixing. Indeed, if the latter flows are mixing, then the usual techniques used to prove the essential value criterion for $S_{\alpha\varphi}$, which is equivalent to ergodicity, fail, and we devise a new method to prove this criterion, which we hope could be useful in tackling other problems of ergodicity for cocycles preserving an infinite measure.

Key words and phrases: Cylindrical cascade, essential value, logarithmic and phrases.

DOI: https://doi.org/10.17323/1609-4514-2006-6-4-657-672

Full text: http://www.ams.org/.../abst6-4-2006.html
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Bibliographic databases:

MSC: 37C40, 37A20, 37C10
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Citation: B. R. Fayad, M. Lemańczy, “On the ergodicity of cylindrical transformations given by the logarithm”, Mosc. Math. J., 6:4 (2006), 657–672

Citation in format AMSBIB
\Bibitem{FayLem06} \by B.~R.~Fayad, M.~Lema{\'n}czy \paper On the ergodicity of cylindrical transformations given by the logarithm \jour Mosc. Math.~J. \yr 2006 \vol 6 \issue 4 \pages 657--672 \mathnet{http://mi.mathnet.ru/mmj264} \crossref{https://doi.org/10.17323/1609-4514-2006-6-4-657-672} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2291157} \zmath{https://zbmath.org/?q=an:1130.37341} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208596000003} 

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• http://mi.mathnet.ru/eng/mmj/v6/i4/p657

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This publication is cited in the following articles:
1. Conze J.-P., Fra̧czek K., “Cocycles over interval exchange transformations and multivalued Hamiltonian flows”, Adv. Math., 226:5 (2011), 4373–4428
2. Fraczek K., Ulcigrai C., “Ergodic Properties of Infinite Extensions of Area-Preserving Flows”, Math. Ann., 354:4 (2012), 1289–1367
3. Cirilo P., Lima Yu., Pujals E., “Law of Large Numbers For Certain Cylinder Flows”, Ergod. Theory Dyn. Syst., 34:3 (2014), 801–825
4. Fayad B., Kanigowski A., “Multiple Mixing For a Class of Conservative Surface Flows”, Invent. Math., 203:2 (2016), 555–614