On the ergodicity of cylindrical transformations given by the logarithm

Abstract

Given $\a \in [0,1]$ and $φ: \T \to \R$ measurable, the {\it cylindircal cascade} $S_{\a,φ}$ is the map from $\T \times \R$ to itself given by $S_{\a,φ} (x,y) = (x+\a,y+φ(x))$ 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 $S_{\a,φ}$ are ergodic for every smooth function $φ$ with a logarithmic singularity, provided that the average of $φ$ vanishes. Closely related to $S_{\a,φ}$ are the special flows constructed above $R_\a$ and under $φ+c$ where $c \in \R$ is such that $φ+c>0$. In the case of a function $φ$ with an asymmetric logarithmic singularity our result gives the first examples of ergodic cascades $S_{\a,φ}$ 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 $S_{\a,φ}$, 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.