Critical functions for complex analytic maps

Critical functions measure the width of the domain of stability around a given fixed point or an invariant circle for complex analytic and area-preserving maps. The author discusses their dependence on the rotation number of the invariant curves and proposes some new methods to determine them based on the existence of critical points and on some properties of quasiconformal maps. By means of the majorant series method some rigorous estimates are given for complex area-preserving maps like the semistandard map and the modulated singular map. In particular, the author makes use of the Brjuno function to interpolate critical maps and proves that the convergence of the Brjuno function is a necessary and sufficient condition for the existence of an analytic invariant curve of a given rotation number. The author also discusses the optimality of the rigorous bounds obtained.