Cantor sets and Lipschitz actions on circles and trees

We investigate the minimal sets of intransitive homeomorphisms of the circle with irrational rotation number, and construct examples of circle homeomorphisms that have any of a class of linear hyperbolic Cantor sets as their minimal sets. The examples constructed have the golden mean as a rotation number and are bi-Lipschitz. We build these examples using a binary tree model of the intervals in the complement of a Cantor set. We use the trees and tree rotations to understand the combinatorics of the action of the circle homeomorphism on the intervals outside of the Cantor set.