Rafael de la Llave, Nikola P. Petrov
Regularity of Conjugacies Between Critical Circle Maps:
An Experimental Study
(2572K, PS)
ABSTRACT. We develop numerical implementations
of several criteria to asses the regularity of functions.
The criteria are
based on 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 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 indeed observable by
numerical computation.
Some universal numbers predicted can indeed be computed
reliably. As a result of our calculations, we obtain that several
simple upper bounds seem to be sharp
in some cases, but not in others. This indicates that there may
be conceptually different mechanisms in play.