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.
Publié le : 2002-05-14
Classification:
Critical circle maps,
self-similarity,
renormalization,
smoothness of conjugacies,
37E10,
37-04,
37Cxx,
37Mxx,
43A99
@article{1062621217,
author = {de la Llave, Rafael and Petrov, Nikola P.},
title = {Regularity of conjugacies between critical circle maps: an experimental study},
journal = {Experiment. Math.},
volume = {11},
number = {3},
year = {2002},
pages = { 219-241},
language = {en},
url = {http://dml.mathdoc.fr/item/1062621217}
}
de la Llave, Rafael; Petrov, Nikola P. Regularity of conjugacies between critical circle maps: an experimental study. Experiment. Math., Tome 11 (2002) no. 3, pp. 219-241. http://gdmltest.u-ga.fr/item/1062621217/