T. M. Seara, J. Villanueva
Asymptotic Behaviour of the Domain of
Analyticity of Invariant Curves of the Standard Map
(173K, LaTeX)
ABSTRACT. In this paper we consider
the standard map, and we study the invariant curve
obtained by analytical continuation,
with respect to the perturbative parameter $\epsilon$,
of the invariant circle of rotation number the golden mean
corresponding to the case $\epsilon=0$. We show that, if we consider the
parameterization that conjugates the dynamics of this curve to an
irrational rotation, the domain of definition of this conjugation
has an asymptotic boundary of analyticity
when $\epsilon\to 0$ (in the sense of the singular perturbation
theory). This boundary is obtained studying the conjugation
problem for the so-called semi-standard map.
To prove this result we have used KAM-like methods adapted to the
framework of singular perturbation theory, as well as matching
techniques to join different pieces of the conjugation,
obtained in different parts of its domain of analyticity.