Birkhoff generic points on curves

Abstract: Let $a_t$ be a diagonal flow on the space X of unimodular lattices in R^n. A point x in X is called Birkhoff generic if a_t.x equidistributes in X as t\to \infty. By Birkhoff ergodic theorem, almost every point x in X is Birkhoff generic. One may ask whether the same is true when the point x is sampled according to a measure singular to Lebesgue. In a joint work with Andreas Wieser, we discuss the case of a generic point x in an analytic curve in X, and show that under certain conditions, it must be Birkhoff generic. This Birkhoff genericity result has various applications in Diophantine approximation. In this talk we will relate Birkhoff genericity to approximations of real numbers by algebraic numbers of degree at most n.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar