Bernoulli property for certain skew products over hyperbolic systems

We study the Bernoulli property for a class of partially hyperbolic systems arising from skew products. More precisely, we consider map $(T,M,\mu )$, where $\mu$ is Gibbs measure, an aperiodic Hölder continuous cocycle $\phi :M\to \mathbb {R}$ with zero mean and zero-entropy flow $(K_t,N,\nu )$. then product \begin{equation*} T_\phi (x,y)=(Tx,K_{\phi (x)}y), \end{equation*} acting on $(M\times N,\mu \times \nu show that if $(K_t)$ slow growth has good equidistribution properties, $T_\phi$ remains Bernoulli. In particular, our main result applies to being typical translation surface genus $\geq 1$ or smooth reparametrization isometric flows $\mathbb {T}^2$. This provides examples non-algebraic, which are center non-isometric (in fact might be weakly mixing).