We prove some ergodic-theoretic rigidity properties of the action of on moduli space. In
particular, we show that any ergodic measure invariant under the action of the upper …

Translation surfaces can be defined in an elementary way via polygons, and arise naturally
in the study of various basic dynamical systems. They can also be defined as di fferentials …

We prove that affine invariant manifolds in strata of flat surfaces are algebraic varieties. The
result is deduced from a generalization of a theorem of Möller. Namely, we prove that the …

Let M be a translation surface. We show that certain deformations of M supported on the set
of all cylinders in a given direction remain in the GL+(2, ℝ)–orbit closure of M. Applications …

The algebraic hull of the Kontsevich–Zorich cocycle Page 1 Annals of Mathematics 188 (2018),
281–313 https://doi.org/10.4007/annals.2018.188.1.5 The algebraic hull of the Kontsevich–Zorich …

We study the boundary of an affine invariant submanifold of a stratum of translation surfaces
in a partial compactification consisting of all finite area Abelian differentials over nodal …

The field of definition of an affine invariant submanifold ℳ is the smallest subfield of ℝ such
that ℳ can be defined in local period coordinates by linear equations with coefficients in this …

We prove that the Birkhoff pointwise ergodic theorem and the Oseledets multiplicative
ergodic theorem hold for every flat surface in almost every direction. The proofs rely on the …

This short expository note gives an elementary introduction to the study of dynamics on
certain moduli spaces and, in particular, the recent breakthrough result of Eskin, Mirzakhani …

A translation surface is a multifaceted object that can be studied with the tools of dynamics,
analysis, or algebraic geometry. Moduli spaces of translation surfaces exhibit equally rich …