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 , μ ) (T,M,\mu ) , where μ \mu is a Gibbs measure, an aperiodic Hölder continuous cocycle ϕ : M → R \phi :M\to \mathbb {R} with zero mean and a zero-entropy flow ( K t , N , ν ) (K_t,N,\nu ) . We then study the skew product T ϕ ( x , y ) = ( T x , K ϕ ( x ) y ) , \begin{equation*} T_\phi (x,y)=(Tx,K_{\phi (x)}y), \end{equation*} acting on ( M × N , μ × ν ) (M\times N,\mu \times \nu ) . We show that if ( K t ) (K_t) is of slow growth and has good equidistribution properties, then T ϕ T_\phi remains Bernoulli. In particular, our main result applies to ( K t ) (K_t) being a typical translation flow on a surface of genus ≥ 1 \geq 1 or a smooth reparametrization of isometric flows on T 2 \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).
Accepted Version
Published Version
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