Are there critical points on the boundaries of singular domains? (Q1072677)

In continuation with the works of M. P. Fatou, C. L. Siegel and E. Ghys, the author proves that if a rational function f of the Riemann sphere of degree not less than two leaves invariant a singular domain C on which the rotation number of f satisfies a diophantine condition, provided that on \(\bar C\) f is injective, then each boundary component of C contains critical points of f. Several applications of the main theorem are pointed out. Furthermore a survey of the theory of iteration of entire functions of \({\mathbb{C}}\) is made.