Backward orbits and petals of semigroups of holomorphic self-maps of the unit disc (1804.09744v1)

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.