Skip to main content Accessibility help
×
Home

Aperiodicity at the boundary of chaos

  • STEVEN HURDER (a1) and ANA RECHTMAN (a2)

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.

Copyright

References

Hide All
[1] Bowen, R.. Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971), 401414.
[2] Edwards, R., Millett, K. and Sullivan, D.. Foliations with all leaves compact. Topology 16 (1977), 1332.
[3] Epstein, D. B. A. and Vogt, E.. A counterexample to the periodic orbit conjecture in codimension 3. Ann. of Math. (2) 108 (1978), 539552.
[4] Ghys, É.. Construction de champs de vecteurs sans orbite périodique (d’après Krystyna Kuperberg), Séminaire Bourbaki, Vol. 1993/94, Exp. No. 785. Astérisque 227 (1995), 283307.
[5] Hurder, S. and Rechtman, A.. The dynamics of generic Kuperberg flows. Astérisque 377 (2016), 1250.
[6] Hurder, S. and Rechtman, A.. Perspectives on Kuperberg flows. Preprint, 2016, arXiv:1607.00731, submitted.
[7] Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137173.
[8] Kuperberg, K.. A smooth counterexample to the Seifert conjecture. Ann. of Math. (2) 140 (1994), 723732.
[9] Kuperberg, G. and Kuperberg, K.. Generalized counterexamples to the Seifert conjecture. Ann. of Math. (2) 144 (1996), 239268.
[10] Matsumoto, S.. K. M. Kuperberg’s C counterexample to the Seifert conjecture. Sūgaku, Math. Soc. Japan 47 (1995), 3845; Translation: Sugaku expositions, Amer. Math. Soc. 11, (1998) 39–49.
[11] Matsumoto, S.. The unique ergodicity of equicontinuous laminations. Hokkaido Math. J. 39 (2010), 389403.
[12] Sullivan, D.. A counterexample to the periodic orbit conjecture. Publ. Math. Inst. Hautes Études Sci. 46 (1976), 514.
[13] Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79) . Springer, New York, 1982.
[14] Wilson, F. W. Jr. On the minimal sets of non-singular vector fields. Ann. of Math. (2) 84 (1966), 529536.

Related content

Powered by UNSILO

Aperiodicity at the boundary of chaos

  • STEVEN HURDER (a1) and ANA RECHTMAN (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.