Renormalization of Aubry-Mather Cantor sets

Abstract

Letf be a two-dimensional area-preserving twist map. Given an irrational rotation number ω in the rotation interval off, there is an invariant recurrent set on whichf preserves the circular ordering and which has rotation number ω. For large nonlinearity, the parameter regime we are interested in, this set is a Cantor set. We show that well-ordered (minimizing) sets with rotation numbers close to ω are exponentially close to the Cantor set under study. The detailed configuration of well-ordered (minimizing) sets is universal and depends on one parameter, namely the Lyapunov coefficient of the Cantor set. There is a quantitative correspondence between this and similar behavior in the noninvertible circle map.