Open Access
Translator Disclaimer
2002 Regularity of conjugacies between critical circle maps: an experimental study
Nikola P. Petrov, Rafael de la Llave
Experiment. Math. 11(2): 219-241 (2002).


We develop numerical implementations of several criteria to assess the regularity of functions. The criteria are based on the finite difference method and harmonic analysis: Littlewood-Paley theory, and wavelet analysis.

As a first application of the methods, we study the regularity of conjugacies between critical circle maps (i.e., differentiable homeomorphisms with a critical point) with a golden mean rotation number. These maps have a very well-developed mathematical theory as well as a wealth of numerical studies.

We compare the results produced by our methods among themselves and with theorems in the mathematical literature. We confirm that several of the features that are predicted by the mathematical results are observable by numerical computation. Some universal numbers predicted can be computed reliably. Our calculations suggest that several simple upper bounds are sharp in some cases, but not in others. This indicates that there may be conceptually different mechanisms at play.


Download Citation

Nikola P. Petrov. Rafael de la Llave. "Regularity of conjugacies between critical circle maps: an experimental study." Experiment. Math. 11 (2) 219 - 241, 2002.


Published: 2002
First available in Project Euclid: 3 September 2003

zbMATH: 1116.37306
MathSciNet: MR1959265

Primary: 37E10
Secondary: 37-04 , 37Cxx , 37Mxx , 43A99

Keywords: critical circle maps , renormalization , self-similarity , smoothness of conjugacies

Rights: Copyright © 2002 A K Peters, Ltd.


Vol.11 • No. 2 • 2002
Back to Top