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

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

Infinitesimal generators and the Loewner equation on complete hyperbolic manifolds

We characterize infinitesimal generators on complete hyperbolic complex manifolds without any regularity assumption on the Kobayashi distance. This allows to prove a general Loewner type equation with regularity of any order $d\in [1,+\infty ]$. Finally, based on these results, we focus on some open problems naturally arising.Comment: 13 pages; misprints corrected and some proofs clarifie

Backward orbits in the unit ball

We show that, if $f\colon \mathbb{B}^q\to \mathbb{B}^q$ is a holomorphic self-map of the unit ball in $\mathbb{C}^q$ and $\zeta\in \partial \mathbb{B}^q$ is a boundary repelling fixed point with dilation $\lambda>1$, then there exists a backward orbit converging to $\zeta$ with step $\log \lambda$. Morever, any two backward orbits converging to the same boundary repelling fixed point stay at finite distance. As a consequence there exists a unique canonical pre-model $(\mathbb{B}^k,\ell, \tau)$ associated with $\zeta$ where $1\leq k\leq q$, $\tau$ is a hyperbolic automorphism of $\mathbb{B}^k$, and whose image $\ell(\mathbb{B}^k)$ is precisely the set of starting points of backward orbits with bounded step converging to $\zeta$. This answers questions in [8] and [3,4].Comment: 9 page