On the ergodicity of cylindrical transformations given by the logarithm

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.