Regularity of critical objects in dynamical systems
University of Oklahoma
IIMAS-UNAM, FENOMEC, Mexico City, Mexico
Last modified: March 29, 2006
We discuss numerical methods we developed for analysing
the Holder regularity of one-dimensional functions based on
harmonic analysis (Littlewood-Paley theory and wavelet analysis).
We illustrate these methods on the example of the regularity
of functions related to the critical invariant circles
of area-preserving twist maps. Such maps are simple models
for the behavior of dynamical systems, and their critical (i.e., just
before destruction) invariant circles are the "last barriers"
to chaotic behavior of the system.
The small-scale properties of these critical objects (such as
their Holder regularity) are of interest in theory of dynamical systems
because they reflect the long-time behavior of the system.