Rates of mixing for the Weil–Petersson geodesic flow: exponential mixing in exceptional moduli Spaces

Abstract

We establish exponential mixing for the geodesic flow $${\varphi_t\colon T^1S\to T^1S}$$ of an incomplete, negatively curved surface S with cusp-like singularities of a prescribed order. As a consequence, we obtain that the Weil–Petersson flows for the moduli spaces $${\mathcal{M}_{1,1}}$$ and $${\mathcal{M}_{0,4}}$$ are exponentially mixing, in sharp contrast to the flows for $${\mathcal{M}_{g,n}}$$ with $${3g-3+n > 1}$$ , which fail to be rapidly mixing. In the proof, we present a new method of analyzing invariant foliations for hyperbolic flows with singularities, based on changing the Riemannian metric on the phase space T 1 S and rescaling the flow $${\varphi_t}$$ .