Regularity of conjugacies between critical circle maps: an experimental study

Abstract

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.