Fourier decay of equilibrium states on hyperbolic surfaces

TL;DR

This paper proves Fourier decay properties of equilibrium states on hyperbolic surfaces, linking geometric, dynamical, and harmonic analysis aspects, with implications for the Fourier dimension of the non-wandering set.

Contribution

It establishes Fourier decay for equilibrium states on hyperbolic surfaces and connects these states to stationary measures with exponential moments under certain conditions.

Findings

Equilibrium states exhibit Fourier decay with power law rates.

Non-wandering sets on hyperbolic surfaces have positive Fourier dimension.

Stationary measures with exponential moments are constructed for certain random walks.

Abstract

Let $\Gamma$ be a (convex-)cocompact group of isometries of the hyperbolic space $\mathbb{H}^d$, let $M := \mathbb{H}^d/\Gamma$ 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 $(\Gamma,F)$-Patterson-Sullivan densities are stationary measures with exponential moment for some random walk on $\Gamma$. 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.