Exponential mixing for the Teichmüller flow

Abstract

Abstract. We study the dynamics of the Teichmüller flow in the moduli space of Abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of Hölder observables. A geometric consequence is that the SL(2, R) action in the moduli space has a spectral gap.

Exponential Decay of Correlations for Piecewise Cone Hyperbolic Contact Flows

“…Moreover, DF (2) m is expressed in terms of γ, Dγ, F (1) m and DF (1) m . All those functions belong to K(C # ) (the first three functions are Lipschitz and bounded, hence in K(C # ), while we proved in Lemma C.4 that…”

“…We proved in (a) that the balls B(Πm, C 1/2 0 ) for m ∈ J ′ are disjoint, therefore all the foliations φ (2) m can be glued together (with the trivial stable foliation outside of m∈J ′ B(Πm, C 1/2 0 )), to get a single foliation parameterised by φ (2) : R d → R d . We emphasize that this new foliation is not necessarily contained in the cone Q( C s ), since the function γ contributes to the derivative of φ (2) .…”

“…We emphasize that this new foliation is not necessarily contained in the cone Q( C s ), since the function γ contributes to the derivative of φ (2) . Nevertheless, it is uniformly transversal to the direction R du × {0} × R, and this will be sufficient for our purposes.…”

“…where F (2) coincides everywhere with a function F (2) m or with the function (x u , x s ) → x u . Since all the derivatives of those functions belong to K(C # ), it follows that DF (2) ∈ K(C # ) (for some other constant C # , worse than the previous one due to the glueing).…”

“…Third step: Pushing the foliation φ (2) with D −1 . This very simple step is the heart of the argument and this is where (C.1) is needed: Define a new foliation by…”