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Aperiodicity at the boundary of chaos

Published online by Cambridge University Press:  04 May 2017

STEVEN HURDER
Affiliation:
Department of Mathematics, University of Illinois at Chicago, 322 SEO (m/c 249), 851 S. Morgan Street, Chicago, IL 60607-7045, USA email hurder@uic.edu
ANA RECHTMAN
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, 04510 Ciudad de México, México email rechtman@im.unam.mx

Abstract

We consider the dynamical properties of $C^{\infty }$-variations of the flow on an aperiodic Kuperberg plug $\mathbb{K}$. Our main result is that there exists a smooth one-parameter family of plugs $\mathbb{K}_{\unicode[STIX]{x1D716}}$ for $\unicode[STIX]{x1D716}\in (-a,a)$ and $a<1$, such that: (1) the plug $\mathbb{K}_{0}=\mathbb{K}$ is a generic Kuperberg plug; (2) for $\unicode[STIX]{x1D716}<0$, the flow in the plug $\mathbb{K}_{\unicode[STIX]{x1D716}}$ has two periodic orbits that bound an invariant cylinder, all other orbits of the flow are wandering, and the flow has topological entropy zero; (3) for $\unicode[STIX]{x1D716}>0$, the flow in the plug $\mathbb{K}_{\unicode[STIX]{x1D716}}$ has positive topological entropy, and an abundance of periodic orbits.

Type
Original Article
Copyright
© Cambridge University Press, 2017 

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