Iterates of holomorphic maps, fixed points, and domains of univalence

Abstract

Fixed points play an important part in the dynamics of a holomorphic map. Given a holomorphic self-map of a unit disc, all of its fixed points, with the exception of at most one of them, lie on the boundary of the disc. Furthermore, it turns out that the existence of an angular derivative and its value at a boundary fixed point affect significantly the behaviour of the map itself and its iterates. In addition, some classical problems in geometric function theory acquire new settings and statements in this context. These questions are considered in this paper. The presentation focuses on the problem of fractional iterations, domains of univalence, and the influence of the angular derivative at a boundary fixed point on the regions of values of Taylor coefficients. Bibliography: 90 titles.