Backward orbits and petals of semigroups of holomorphic self-maps of the unit disc

TL;DR

This paper investigates the structure of backward invariant sets, called petals, in semigroups of holomorphic self-maps of the unit disc, revealing their geometric properties and relations to fixed points.

Contribution

It characterizes petals in terms of fixed points, boundary behavior, and models, providing new insights into their geometric and analytic structure.

Findings

Hyperbolic petals correspond to repelling fixed points.

Parabolic semigroups can have parabolic petals.

Petals are generally Jordan domains with connected boundaries.

Abstract

We study the backward invariant set of one-parameter semigroups of holomorphic self-maps of the unit disc. Such a set is foliated in maximal invariant curves and its open connected components are petals, which are, in fact, images of Poggi-Corradini's type pre-models. Hyperbolic petals are in one-to-one correspondence with repelling fixed points, while only parabolic semigroups can have parabolic petals. Petals have locally connected boundaries and, except a very particular case, they are indeed Jordan domains. The boundary of a petal contains the Denjoy-Wolff point and, except such a fixed point, the closure of a petal contains either no other boundary fixed point or a unique repelling fixed point. We also describe petals in terms of geometric and analytic behavior of K\"onigs functions using divergence rate and universality of models. Moreover, we construct a semigroup having a repelling fixed point in such a way that the intertwining map of the pre-model is not regular.