Nonwandering structures at the period-doubling limit in dimensions $2$ and $3$

Abstract:

A Cantor set supporting an adding machine is the simplest nonwandering structure that can occur at the conclusion of a sequence of perioddoubling bifurcations of plane homeomorphisms. In some families this structure is persistent. In this manuscript it is shown that no plane homeomorphism has nonwandering Knaster continua on which the homeomorphism is semiconjugate to the adding machine. Using a theorem of M. Brown, a three-space homeomorphism is constructed which has an invariant set, $\Lambda$, the product of a Knaster continuum and a Cantor set. $\Lambda$ is chainable, supports positive entropy but contains only power-of-two periodic orbits. And the homeomorphism restricted to $\Lambda$ is semiconjugate to the adding machine. Lastly, a zero topological entropy ${C^\infty }$ disk diffeomorphism is constructed which has large nonwandering structures over a generalized adding machine on a Cantor set.