The stable subset of a univalent self-map

We give a complete description of the stable subset (the union of all backward orbit with bounded step) and of the pre-models of a univalent self-map $f: X\to X$, where $X$ is a Kobayashi hyperbolic cocompact complex manifold, such as the ball or the polydisc in $C^q$. The result is obtained studying the complex structure of a decreasing intersection of complex manifolds, all biholomorphic to $X$

Valiron and Abel equations for holomorphic self-maps of the polydisc

We introduce a notion of hyperbolicity and parabolicity for a holomorphic self-map $f: \Delta^N \to \Delta^N$ of the polydisc which does not admit fixed points in $\Delta^N$. We generalize to the polydisc two classical one-variable results: we solve the Valiron equation for a hyperbolic $f$ and the Abel equation for a parabolic nonzero-step $f$. This is done by studying the canonical Kobayashi hyperbolic semi-model of $f$ and by obtaining a normal form for the automorphisms of the polydisc. In the case of the Valiron equation we also describe the space of all solutions.Comment: A few references are adde

Loewner equations on complete hyperbolic domains

We prove that, on a complete hyperbolic domain D\subset C^q, any Loewner PDE associated with a Herglotz vector field of the form H(z,t)=A(z)+O(|z|^2), where the eigenvalues of A have strictly negative real part, admits a solution given by a family of univalent mappings (f_t: D\to C^q) such that the union of the images f_t(D) is the whole C^q. If no real resonance occurs among the eigenvalues of A, then the family (e^{At}\circ f_t) is uniformly bounded in a neighborhood of the origin. We also give a generalization of Pommerenke's univalence criterion on complete hyperbolic domains.Comment: 19 pages, revised exposition, improved results, added reference

Abstract basins of attraction

Abstract basins appear naturally in different areas of several complex variables. In this survey we want to describe three different topics in which they play an important role, leading to interesting open problems

Simultaneous models for commuting holomorphic self-maps of the ball

We prove that a finite family of commuting holomorphic self-maps of the unit ball $\mathbb{B}^q\subset \mathbb{C}^q$ admits a simultaneous holomorphic conjugacy to a family of commuting automorphisms of a possibly lower dimensional ball, and that such conjugacy satisfies a universal property. As an application we describe when a hyperbolic and a parabolic holomorphic self-map of $\mathbb{B}^q$ can commute.Comment: Final version, to appear on Adv. Mat

Teoremi dei Residui

Si presenta un procedimento per localizzare classi caratteristiche di fibrati nelle singolarita' di opportuni oggetti geometrici e e per ottenere teoremi dei residui

Iron homeostasis in health and disease

Iron is required for the survival of most organisms, including bacteria, plants, and humans. Its homeostasis in mammals must be fine-tuned to avoid iron deficiency with a reduced oxygen transport and diminished activity of Fe-dependent enzymes, and also iron excess that may catalyze the formation of highly reactive hydroxyl radicals, oxidative stress, and programmed cell death. The advance in understanding the main players and mechanisms involved in iron regulation significantly improved since the discovery of genes responsible for hemochromatosis, the IRE/IRPs machinery, and the hepcidin-ferroportin axis. This review provides an update on the molecular mechanisms regulating cellular and systemic Fe homeostasis and their roles in pathophysiologic conditions that involve alterations of iron metabolism, and provides novel therapeutic strategies to prevent the deleterious effect of its deficiency/overload