Bernoullicity of some skew products with hyperbolic base and Kochergin flow in the fiber

We study the Bernoulli property for skew products with hyperbolic diffeomorphisms equipped with a Gibbs measure in the base and Kochergin flows in the fiber, when the cocycle is aperiodic and of zero mean. The flow in the fiber can be represented as a special flow over an irrational rotation and a roof function with power singularity. We show that if the growth near the singularity is given by an exponent smaller than $\frac{1}{2}$, then for almost every rotation the resulting skew product is Bernoulli.