Hamiltonian flows for pseudo-Anosov mapping classes

Abstract

For a given pseudo-Anosov homeomorphism $\varphi$ of a closed surface $S$, the action of $\varphi$ on the Teichm\"uller space $\mathcal T(S)$ preserves the Weil-Petersson symplectic form. We give explicit formulae for two invariant functions $\mathcal T(S)\to \mathbb R$ whose symplectic gradients generate autonomous Hamiltonian flows that coincide with the action of $\varphi$ at time one. We compute the Poisson bracket between these two functions. This amounts to computing the variation of length of a H\"older cocyle on one lamination along a shear vector field defined by another. For a measurably generic set of laminations, we prove that the variation of length is expressed as the cosine of the angle between the two laminations integrated against the product H\"older distribution, generalizing a result of Kerckhoff. We also obtain rates of convergence for the supports of germs of differentiable paths of measured laminations in the Hausdorff metric on a hyperbolic surface, which may be of independent interest.