Periodicity generated by adding machines

Abstract

We show that a homeomorphism of the plane $\mathbb{R}^2$ with an invariant Cantor set $\mathbf {C}$, on which the homeomorphism acts as an adding machine, possesses periodic points arbitrarily close to $\mathbf {C}$. The existence of periodic points near an invariant Cantor set is related to a shape theory question whether a solenoid invariant in a flow defined on $\mathbb{R}^3$ must be contained in a larger movable invariant compactum.