Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T17:17:43.963Z Has data issue: false hasContentIssue false
  • English
  • Français

Ziggurat fringes are self-similar

Published online by Cambridge University Press:  28 December 2015

SUBHADIP CHOWDHURY
SUBHADIP CHOWDHURY*
Affiliation:
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA email subhadip@math.uchicago.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We give explicit formulae for fringe lengths of the Calegari–Walker ziggurats—i.e., graphs of extremal rotation numbers associated with positive words in free groups. These formulae reveal (partial) integral projective self-similarity in ziggurat fringes, which are low-dimensional projections of characteristic polyhedra on the bounded cohomology of free groups. This explains phenomena observed experimentally by Gordenko, Calegari and Walker.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 3 , May 2017 , pp. 739 - 757
Copyright
© Cambridge University Press, 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agnihotri, S. and Woodward, C.. Eigenvalues of products of unitary matrices and quantum Schubert calculus. Math. Res. Lett. 5(6) (1998), 817–836.Google Scholar
Belkale, P.. Local systems on P 1 - S for S a finite set. Compos. Math. 129(1) (2001), 67–86.Google Scholar
Bucher, M., Frigerio, R. and Hartnick, T.. A note on semi-conjugacy for circle actions. Preprint, 2014, arXiv:1410.8350.Google Scholar
Calegari, D. and Walker, A.. Ziggurats and rotation numbers. J. Mod. Dyn. 5(4) (2011), 711–746.Google Scholar
Ecke, R. E., Doyne Farmer, J. and Umberger, D. K.. Scaling of the Arnol’d tongues. Nonlinearity 2(2) (1989), 175–196.Google Scholar
Ghys, É.. Groups acting on the circle. Enseign. Math. (2) 47 (2001), 329–407.Google Scholar
Gordenko, A.. Self-similarity of Jankins–Neumann ziggurat. Preprint, 2015, arXiv:1503.03114.Google Scholar
Jankins, M. and Neumann, W.. Rotation numbers of products of circle homeomorphisms. Math. Ann. 271(3) (1985), 381–400.Google Scholar
Kaplan, S.. Application of programs with maximin objective functions to problems of optimal resource allocation. Oper. Res. 22(4) (1974), 802–807.Google Scholar
Naimi, R.. Foliations transverse to fibers of Seifert manifolds. Comment. Math. Helv. 69(1) (1994), 155–162.Google Scholar