From rational billiards to dynamics on moduli spaces

Abstract

Abstract. 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, and Mohammadi. We also discuss the context and applications of this result, and its connections to other areas of mathematics, such as algebraic geometry, Teichmüller theory, and ergodic theory on homogeneous spaces.

WHAT IS...an Ergodic Transformation?

“…We conclude with a topic that appears in A. Eskin's invited address (see page 17). Recently Eskin, Mirzakhani, and Mohammadi proved a series of remarkable theorems that can be thought of as analogues of Ratner's results in the case of actions of the group of 2 × 2 real matrices with determinant 1 on the moduli space of translation surfaces (see [2]). Translation surfaces were introduced to study billiard flows.…”

Book Review: Twisted Teichmüller curves

“…A decade after McMullen's [54] treatment of the genus 2 case, Eskin and Mirzakhani [17] show that finite ergodic SL 2 (R)-invariant measures are of Lebesgue class and supported on affine varieties; Eskin, Mirzakhani, and Mohammadi [18] show that the closures of SL 2 (R)-orbits are affine manifolds; and Filip [21,22] shows that these closures are algebraic varieties defined over number fields, thus generalizing the aforementioned results for Teichmüller curves. For more on these results and the flurry of work they have inspired see [82]. Finiteness results for primitive Teichmüller curves in certain strata based on these results include those of [47] and [45].…”

The limit set of non-orientable mapping class groups

“…Definition 2.3 (Dianalytic quadratic differential (adapted from [Wri15])). A dianalytic quadratic differential is the quotient of a collection of polygons in C, modulo certain equivalences.…”