Journal of Modern Dynamics (JMD)

Ziggurats and rotation numbers

Pages: 711 - 746, Issue 4, October 2011      doi:10.3934/jmd.2011.5.711

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Danny Calegari - DPMMS, University of Cambridge, Cambridge CB3 0WA, United Kingdom (email)
Alden Walker - Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, United States (email)

Abstract: We establish the existence of new rigidity and rationality phenomena in the theory of nonabelian group actions on the circle and introduce tools to translate questions about the existence of actions with prescribed dynamics into finite combinatorics. A special case of our theory gives a very short new proof of Naimi's theorem (i.e., the conjecture of Jankins-Neumann) which was the last step in the classification of taut foliations of Seifert fibered spaces.

Keywords:  Rotation number, ziggurat, rationality, Arnol’d tongues, rigidity, circle homeomorphism, 1-dimensional dynamics.
Mathematics Subject Classification:  Primary: 37E45, 58D05; Secondary: 58F10.

Received: October 2011;      Revised: November 2011;      Available Online: March 2012.