We establish bounds on the minimal asymptotic pseudo-Anosov translation lengths on the complex of curves of orientable surfaces. In particular, for a closed surface with genus g > 2, we show that there are positive constants a 1 < a 2 such that the minimal translation length is bounded below and above by a 1 =g 2 and a 2 =g 2 .

A natural generalization of interval exchange maps are linear involutions, first introduced by Danthony and Nogueira [Measured foliations on non-orientable surfaces.Ann. Sci. Éc. Norm. Supér.(4)26(6) (1993), 645–664]. Recurrent train tracks with a single switch provide a subclass of linear involutions. We call such linear involutions non-classical interval exchanges. They are related to measured foliations on orientable flat surfaces. Non-classical interval exchanges can be studied as a dynamical system by considering Rauzy induction in this context. This gives a refinement process on the parameter space similar to Kerckhoff’s simplicial systems. We show that the refinement process gives an expansion that has a key dynamical property calleduniform distortion. We use uniform distortion to prove normality of the expansion. Consequently, we prove an analog of Keane’s conjecture: almost every non-classical interval exchange is uniquely ergodic. Uniform distortion has been independently shown in [A. Avila and M. Resende. Exponential mixing for the Teichmüller flow in the space of quadratic differentials, http://arxiv.org/abs/0908.1102].

Given a measure on the Thurston boundary of Teichmüller space, one can pick a geodesic ray joining some basepoint to a randomly chosen point on the boundary. Different choices of measures may yield typical geodesics with different geometric properties. In particular, we consider two families of measures: the ones which belong to the Lebesgue or visual measure class, and harmonic measures for random walks on the mapping class group generated by a distribution with finite first moment in the word metric. We consider the ratio between the word metric and the relative metric of approximating mapping class group elements along a geodesic ray, and prove that this ratio tends to infinity along almost all geodesics with respect to Lebesgue measure, while the limit is finite along almost all geodesics with respect to harmonic measure. As a corollary, we establish singularity of harmonic measure. We show the same result for cofinite volume Fuchsian groups with cusps. As an application, we answer a question of Deroin-Kleptsyn-Navas about the vanishing of the Lyapunov expansion exponent.

Abstract. We determine the asymptotic behavior of the optimal Lipschitz constant for the systole map from Teichmüller space to the curve complex.

For a non-uniform lattice in SL(2, R), we consider excursions in cusp neighborhoods of a random geodesic on the corresponding finite area hyperbolic surface or orbifold. We prove a strong law for a certain partial sum involving these excursions. This generalizes a theorem of Diamond and Vaaler for continued fractions [9]. In the Teichmüller setting, we consider invariant measures for the SL(2, R) action on the moduli spaces of quadratic differentials. By the work of Eskin and Mirzakhani [12], these measures are supported on affine invariant submanifolds of a stratum of quadratic differentials. For a Teichmüller geodesic random with respect to a SL(2, R)-invariant measure, we study its excursions in thin parts of the associated affine invariant submanifold. Under a regularity hypothesis for the invariant measure, we prove similar strong laws for certain partial sums involving these excursions. The limits in these laws are related to the volume asymptotic of the thin parts. By Siegel-Veech theory, these are given by various Siegel-Veech constants. As a direct consequence, we show that the word metric grows faster than T log T along Teichmüller geodesics random with respect to the Masur-Veech measure.2010 Mathematics Subject Classification. 30F60, 32G15.