Boundary dynamics in unbounded Fatou components

Abstract

We study the behaviour of a transcendental entire map f : C-+ C on an unbounded invariant Fatou component U, assuming that infinity is accessible from U. It is well-known that U is simply connected. Hence, by means of a Riemann map phi: D-+ U and the associated inner function g := phi-1 degrees f degrees phi, the boundary of U is described topologically in terms of the disjoint union of clusters sets, each of them consisting of one or two connected components in C, improving the results in (see Baker and Dom & imath;nguez [Ann. Acad. Sci. Fenn. Math. 24 (1999), pp. 437-464]; Bargmann [Cambridge Univ. Press, Cambridge, 2008]). Moreover, under mild assumptions on the location of singular values in U (allowing them even to accumulate at infinity, as long as they accumulate through accesses to infinity), we show that periodic and escaping boundary points are dense in partial derivative U, and that all periodic boundary points accessible from U. Finally, under similar conditions, the set of singularities of g is shown to have zero Lebesgue measure, strengthening substantially the results (see Evdoridou et al. [J. Math. Anal. Appl. 477 (2019), pp. 536-550; Arnold Math. J. 6 (2020), pp. 459-493]).