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