The Bernoulli property for $PS{L}_2\mathbb{R}$ skew products over exponentially mixing base

We study the Bernoulli property for skew products of the form $f(x,y)=(g(x),A(x)y)$ on $M\times PS{L}_2(\mathbb{R})\mathrm{/}\mathrm{\Gamma }$, where $g$ is a smooth, measure-preserving, exponentially mixing diffeomorphism, $\mathrm{\Gamma }<PS{L}_2(\mathbb{R})$ is a cocompact lattice, and $A\text{\hspace{0.17em}⁣}:M\to {\mathrm{PSL}}_2(\mathbb{R})$ is a $C^{1+\alpha }$ cocycle. We prove that if $A$ has a nonzero Lyapunov exponent, then $f$ is Bernoulli. The proof does not require that $f$ be exponentially mixing. This provides a new method for establishing the Bernoulli property, which applies to a large class of skew products.