Arithmetic and dynamics on varieties of Markoff type

Abstract

The Markoff equation $x^2+y^2+z^2=3xyz$, which arose in his spectacular thesis in 1879, is ubiquitous in a tremendous variety of contexts. After reviewing some of these, we discuss Hasse principle, asymptotics of integer points, and, in particular, recent progress towards establishing forms of strong approximation on varieties of Markoff type, as well as ensuing implications, diophantine and dynamical.