On the ergodicity of cylindrical transformations given by the logarithm

Abstract

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