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Holomorphic Evolution Equations

Final Report Summary - HEVO (Holomorphic Evolution Equations)

The ERC project HEVO dealt with holomorphic evolution equations, namely, those dynamical systems with holomorphic data which arise from autonomous and non-autonomous semi-complete vector fields. Examples of such systems can be naturally find in nature. It also dealt with the related topics of discrete local and semi-local iteration of holomorphic maps. The main focus was to extend the classical one-dimensional Loewner theory to a general theory which holds on complex manifolds and to use it to investigate the dynamics, the geometrical properties and boundary behavior of univalent mappings in one and several complex variables, especially in domains such as the unit ball of the complex space. In this context, it was proved the existence and essential uniqueness of the solution to the Loewner PDE equation in higher dimension, showing also that, if the starting equation is defined in complete hyperbolic starlike domains, the associated Loewner chain has values in the complex space. New variations of the Loewner chains have been discovered and applied in order to study the class of univalent mappings having parametric representation. In this context, it was found an unexpected example of a bounded support functions and many new phenomena for univalent mappings, completely different from the classical one-dimensional case, were found, opening the interesting problem of a “multi-dimensional” Bieberbach conjecture. The construction exploited for such results is a “categorial analysis”, known as abstract basins of attraction—-a construction pretty much related to the Bedford conjecture—-which allowed also to model holomorphic self-maps of complex manifolds and solve with hyperbolic base spaces the Schroeder, Abel and Valiron functional equations. Boundary behavior of holomorphic objects such as infinitesimal generators and maps has been successfully studied. In particular, a “boundary open mapping theorem” for holomorphic mappings between bounded strongly pseudo convex domains was established and the jet description of infinitesimal generators in the unit ball has been done. A new Carathèodory prime ends type theory, the "horosphere topology" has been introduced and used to get extensions of univalent maps to the boundary. Boundary regular fixed points for semigroups have been investigated in strongly convex domains. In the one dimensional case, all boundary singularities of infinitesimal generators (the so-called “fractional singularities”) have been classified and characterized both in terms of the shape of the corresponding Koenigs function and the dynamics of the associated semigroup. Relations with the Brennan conjecture and beta-numbers were found. A precise characterization of boundary regular fixed points for evolution families and Herglotz vector fields was also given.
Precise growth estimates and Abel iterative renormalization for infinitesimal generators and more general pseudo-contractive maps (a newly introduced class of maps which contain both unbounded linear operators and holomorphic maps) were found in Banach spaces. Finally, from the side of holomorphic dynamics, the dynamics of germs of holomorphic mappings with resonances, has been fully described, providing parabolic basins of attraction and Fatou coordinates for non-parabolic germs in higher dimension and examples of automorphisms of the two dimensional complex space with a non-simply connected basin of attraction. Transcendental dynamics and the density of hyperbolicity conjecture were also investigated.