Bers and Hénon, Painlevé and Schrödinger

Abstract

In this article, we pursue the study of the holomorphic dynamics of mapping class groups on two-dimensional character varieties, also called trace-map dynamics in the literature, as initiated in [44] (see also [20]). We show that the dynamics of pseudo-Anosov mapping classes resembles in many ways the dynamics of Hénon mappings, and then we apply this idea to answer open questions concerning

(1) the geometry of discrete and faithful representations of free groups into $\mathrm{SL}(2,C),$

(2) the dynamics of Painlevé sixth equations, and

(3) the spectrum of certain discrete Schrödinger operators