On ergodicity of some cylinder flows

We study ergodicity of cylinder flows of the form   ${\displaystyle T_f:\left\{symT\right\}\times ℝ\to \left\{symT\right\}\times ℝ}$, ${\displaystyle T_{f(x,y)}=(x+\alpha ,y+f\left(x\right))}$, where ${\displaystyle f:\left\{symT\right\}\to ℝ}$ 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 ${\displaystyle D^{k}f}$ is piecewise absolutely continuous (but not continuous) with zero sum of jumps. We show that if the points of discontinuity of ${\displaystyle D^{k}f}$ have some good properties, then ${\displaystyle T_f}$ is ergodic. Moreover, there exists ${\displaystyle {\epsilon }_f>0}$ such that if ${\displaystyle v:\left\{symT\right\}\to ℝ}$ is a function with zero integral such that ${\displaystyle D^{k}v}$ is of bounded variation with ${\displaystyle Var\left(D^{k}v\right)<{\epsilon }_f}$, then ${\displaystyle T_{f+v}}$ is ergodic.