Anisotropic spaces for the bilateral shift

Given two Hölder potentials ${\phi }_{+}$ and ${\psi }_{-}$ for the unilateral shift, we define anisotropic Banach spaces of distributions on the bilateral shift space with a finite alphabet. On these spaces, the transfer operator for the bilateral shift is quasicompact with a spectral gap, and the unique Gibbs state associated with ${\phi }_{+}$ spans its $1$-eigenspace. This result allows us to establish exponential decay of correlations for Hölder observables and a wide range of measures on the bilateral shift space.