Rigidity of Holomorphic Generators of One-Parameter Semigroups and a Non-Autonomous Denjoy-Wolff Theorem

Main topic of the first part of this work is an investigation about rigidity phenomena of infinitesimal generators of one-parameter semigroups of holomorphic self-maps of domains of \(\mathbb{C}^{n}\). By a rigidity condition we mean a sufficient condition that forces an infinitesimal generator to identically vanish. We start describing discrete iteration theory of the unit disc, just to put rational iteration in a proper context. Then, after a presentation of known rigidity results in the unit disc and the unit ball, we present our main results for strongly convex domains of \(\mathbb{C}^{n}\), also providing some new proofs of already known results. Then we move to non-autonomous holomorphic dynamical systems in the unit disc, and we focus on evolution families. After presenting the relevant definitions and properties, we extend, to some extent, the classical Denjoy-Wolff Theorem to evolution families: we show that here the dynamical landscape is reacher then for discrete or rational iteration.