Arturo Olvera, Nikola P. Petrov
Regularity properties of critical invariant circles of twist maps, and their universality
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ABSTRACT. We compute accurately the golden critical invariant circles
of several area-preserving twist maps of the cylinder.
We define some functions related to the invariant
circle and to the dynamics of the map restricted to the circle
(for example, the conjugacy between the circle map
giving the dynamics on the invariant circle
and a rigid rotation on the circle).
The global Holder regularities of these functions are low
(some of them are not even once differentiable).
We present several conjectures about the universality
of the regularity properties of the critical circles
and the related functions.
Using a Fourier analysis method
developed by R. de la Llave and one of the authors,
we compute numerically the Holder regularities of these functions.
Our computations show that -- withing their numerical accuracy --
these regularities are the same for the different maps studied.
We discuss how our findings are related to some previous results:
(a) to the constants giving the scaling behavior of the iterates
on the critical invariant circle (discovered by Kadanoff and Shenker);
(b) to some characteristics of the singular invariant measures
connected with the distribution of iterates.
Some of the functions studied have pointwise Holder
regularity that is different at different points.
Our results give a convincing numerical support
to the fact that the points with different Holder exponents
of these functions are interspersed in the same way for
different maps, which is a strong indication
that the underlying twist maps belong to the same universality class.