Bernoulli property for certain skew products over hyperbolic systems

Abstract:

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 Hölder 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 \begin{equation*} T_\phi (x,y)=(Tx,K_{\phi (x)}y), \end{equation*} 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).