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 $\varphi :M\to \mathbb{R}$ with zero mean and a zero-entropy flow $(K_t,N,\nu )$. We then study the skew product $$T_{\varphi }(x,y)=(Tx,K_{\varphi (x)}y),$$ acting on $(M\times N,\mu \times \nu )$. We show that if $(K_t)$ is of slow growth and has good equidistribution properties, then $T_{\varphi }$ remains Bernoulli. In particular, our main result applies to $(K_t)$ being a typical translation flow on a surface of genus $\ge 1$ 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).

References in corpus (2)

• On the non-equivalence of the Bernoulli and K properties in dimension four
• Scenery entropy as an invariant of RWRS processes

Cited by in corpus (1)

• Exponential mixing implies Bernoulli