Fourier decay of equilibrium states on hyperbolic surfaces

Let $\mathrm{\Gamma }$ be a (convex-)cocompact group of isometries of the hyperbolic space ${\mathbb{H}}^d$, let $M:={\mathbb{H}}^d\mathrm{/}\mathrm{\Gamma }$ be the associated hyperbolic manifold, and consider a real valued potential $F$ on its unit tangent bundle $T^{1}M$. Under a natural regularity condition on $F$, we prove that the associated $(\mathrm{\Gamma },F)$-Patterson-Sullivan densities are stationary measures with exponential moment for some random walk on $\mathrm{\Gamma }$. As a consequence, when $M$ is a surface, the associated equilibrium state for the geodesic flow on $T^{1}M$ exhibit “Fourier decay”, in the sense that a large class of oscillatory integrals involving it satisfies power decay. It follows that the non-wandering set of the geodesic flow on convex-cocompact hyperbolic surfaces has positive Fourier dimension, in a sense made precise in the appendix.