Rates of mixing for the Weil-Petersson geodesic flow II: exponential mixing in exceptional moduli spaces

Abstract

We establish exponential mixing for the geodesic flow ${\phi }_t:T^{1}S\to T^{1}S$ 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 ${\phi }_t$.

42 pages

References in corpus (2)

• Exponential Decay of Correlations for Finite Horizon Sinai Billiard Flows
• Teichmueller flow and Weil-Petersson flow

Cited by in corpus (1)

• Polynomial decay of correlations for flows, including Lorentz gas examples