Event № 63
In a recent paper with L. Arosio (accepted in Trans. Amer. Math. Soc.) we introduced a universal way to construct a universal hyperbolic semi-model for every univalent self-map of the unit ball (that is, a holomorphic semiconjugation to a possibly lower dimensional ball which has the property that every other semi-conjugation to a hyperbolic space factorizes through this). This result has been very recently extended to not necessarily univalent mappings by L. Arosio. Let k be the dimension of the base space of the model for a holomorphic self-map f of the unit ball. Then (in a work in progress with L. Arosio) we proved that the map f has an f-absorbing open domain in the ball on which the rank of f is at least k. In case k=the dimension of the ball where f is defined, we proved that for each orbit the map f is eventually univalent on any given Kobayashi ball. These results can be seen as a generalization of some results of Pommerenke for the unit disc. Using such results, we can prove that every holomorphic map of the unit ball commuting with a hyperbolic self-map for which k is maximal, has to be either hyperbolic or has to fix at least a slice containing the Denjoy-Wolff point of f. When k< maximal dimension, there are examples of hyperbolic maps commuting with parabolic maps (in fact, semigroups). The aim of this talk is to explain these results.
Please note the unusual day, time and place!