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Filippo Bracci

Publications and source records attributed to Filippo Bracci.

At least 55 records · Page 3Linked to original sources
arXiv2015

Invariant holomorphic foliations on Kobayashi hyperbolic homogeneous manifolds

Let $M$ be a Kobayashi hyperbolic homogenous manifold. Let $\mathcal F$ be a holomorphic foliation on $M$ invariant under a transitive group $G$ of biholomorphisms. We prove that the leaves of $\mathcal F$ are the fibers of a holomorphic $G$-equivariant submersion $π\colon M \to N$ onto a $G$-homogeneous complex manifold $N$. We also show that if $\mathcal Q$ is an automorphism family of a hyperbolic convex (possibly unbounded) domain $D$ in $\mathbb C^n$, then the fixed point set of $\mathcal Q$ is either empty or a connected complex submanifold of $D$.

math.CV↗
arXiv2015

Variation of Loewner chains, extreme and support points in the class $S^0$ in higher dimensions

We introduce a family of natural normalized Loewner chains in the unit ball, which we call "geräumig"---spacious---which allow to construct, by means of suitable variations, other normalized Loewner chains which coincide with the given ones from a certain time on. We apply our construction to the study of support points, extreme points and time-$\log M$-reachable functions in the class $S^0$ of mappings admitting parametric representation.

math.CV↗
arXiv2015

Localized intersection of currents and the Lefschetz coincidence point theorem

We introduce the notion of a Thom class of a current and define the localized intersection of currents. In particular we consider the situation where we have a smooth map of manifolds and study localized intersections of the source manifold and currents on the target manifold. We then obtain a residue theorem on the source manifold and give explicit formulas for the residues in some cases. These are applied to the problem of coincidence points of two maps. We define the global and local coincidence homology classes and indices. A representation of the Thom class of the graph as a Cech-de~Rham cocycle immediately gives us an explicit expression of the index at an isolated coincidence point, which in turn gives explicit coincidence classes in some non-isolated components. Combining these, we have a general coincidence point theorem including the one by S. Lefschetz.

math.CV↗
arXiv2014

Canonical Models for holomorphic iteration

We construct canonical intertwining semi-models with Kobayashi hyperbolic base space for holomorphic self-maps of complex manifolds which are univalent on some absorbing cocompact hyperbolic domain. In particular, in the unit ball we solve the Valiron equation for hyperbolic univalent self-maps and for hyperbolic semigroups.

math.CV↗
arXiv2014

Common boundary regular fixed points for holomorphic semigroups in strongly convex domains

Let $D$ be a bounded strongly convex domain with smooth boundary in $\mathbb C^N$. Let $(ϕ_t)$ be a continuous semigroup of holomorphic self-maps of $D$. We prove that if $p\in \partial D$ is an isolated boundary regular fixed point for $ϕ_{t_0}$ for some $t_0>0$, then $p$ is a boundary regular fixed point for $ϕ_t$ for all $t\geq 0$. Along the way we also study backward iteration sequences for elliptic holomorphic self-maps of $D$.

math.CV↗
arXiv2014

Shearing process and an example of a bounded support function in $S^0(\mathbb B^2)$

We introduce a process, that we call "shearing", which for any given normal Loewner chain produces a normal Loewner chain made of shears automorphisms. As an application, and in stringent contrast to the one-dimensional case, we prove the existence of a starlike bounded function in the class $S^0$ of the ball $\mathbb B^2$ (in fact the restriction of a shear automorphism of $\mathbb C^2$) which is a support point for a linear continuous functional.

math.CV↗
arXiv2013

Abel averages and holomorphically pseudo-contractive maps in Banach spaces

A class of maps in a complex Banach space is studied, which includes both unbounded linear operators and nonlinear holomorphic maps. The defining property, which we call {\sl pseudo-contractivity}, is introduced by means of the Abel averages of such maps. We show that the studied maps are dissipative in the spirit of the classical Lumer-Phillips theorem. For pseudo-contractive holomorphic maps, we establish the power convergence of the Abel averages to holomorphic retractions.

math.CV↗
arXiv2013

Contact points and fractional singularities for semigroups of holomorphic self-maps in the unit disc

We study boundary singularities which can appear for infinitesimal generators of one-parameter semigroups of holomorphic self-maps in the unit disc. We introduce "regular" fractional singularities and characterize them in terms of the behavior of the associated semigroups and Koenigs functions. We also provide necessary and sufficient geometric criteria on the shape of the image of the Koenigs function for having such singularities. In order to do this, we study contact points of semigroups and prove that any contact (not fixed) point of a one-parameter semigroup corresponds to a maximal arc on the boundary to which the associated infinitesimal generator extends holomorphically as a vector field tangent to this arc.

math.CV↗
arXiv2013

Classical and stochastic Löwner-Kufarev equations

In this paper we present a historical and scientific account of the development of the theory of the Löwner-Kufarev classical and stochastic equations spanning the 90-year period from the seminal paper by K.Löwner in 1923 to recent generalizations and stochastic versions and their relations to conformal field theory.

math.CV↗
arXiv2013

Embedding univalent functions in filtering Loewner chains in higher dimension

We discuss the problem of embedding univalent functions into Loewner chains in higher dimension. In particular, we prove that a normalized univalent map of the ball in $\C^n$ whose image is a smooth strongly pseudoconvex domain is embeddable into a normalized Loewner chain (satisfying also some extra regularity properties) if and only if the closure of the image is polynomially convex.

math.CV↗
arXiv2013

Growth estimates for pseudo-dissipative holomorphic maps in Banach spaces

In this paper we introduce a class of pseudo-dissipative holomorphic maps which contains, in particular, the class of infinitesimal generators of semigroups of holomorphic maps on the unit ball of a complex Banach space. We give a growth estimate for maps of this class. In particular, it follows that pseudo-dissipative maps on the unit ball of (infinite-dimensional) Banach spaces are bounded on each domain strictly contained inside the ball. We also present some applications.

math.CV↗
arXiv2013

Boundary regular fixed points in Loewner theory

We characterize regular fixed points of evolution families in terms of analytical properties of the associated Herglotz vector fields and geometrical properties of the associated Loewner chains. We present several examples showing the rôle of the given conditions. Moreover, we study the relations between evolution families and Herglotz vector fields at regular contact points and prove an embedding result for univalent self-maps of the unit disc with a given boundary regular fixed point into an evolution family with prescribed boundary data.

math.CV↗
arXiv2012

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.

math.CV↗
arXiv2012

The range of holomorphic maps at boundary points

We prove a boundary version of the open mapping theorem for holomorphic maps between strongly pseudoconvex domains. That is, we prove that the local image of a holomorphic map $f:D\to D'$ close to a boundary regular contact point $p\in \de D$ where the Jacobian is bounded from zero along normal non-tangential directions has to eventually contain every cone (and more generally every admissible region) with vertex at $f(p)$.

math.CV↗
arXiv2012

Boundary behavior of infinitesimal generators in the unit ball

We prove a Julia-Wolff-Caratheodory type theorem for infinitesimal generators on the unit ball in C^n. Moreover, we study jets expansions at the boundary and give necessary and sufficient conditions on such jets for an infinitesimal generator to generate a group of automorphisms of the ball.

math.CV↗
arXiv2012

Dynamics of quasi-parabolic one-resonant biholomorphisms

In this paper we study the dynamics of germs of quasi-parabolic one-resonant biholomorphisms of $\C^{n+1}$ fixing the origin, namely, those germs whose differential at the origin has one eigenvalue 1 and the others having a one dimensional family of resonant relations. We define some invariants and give conditions which ensure the existence of attracting domains for such maps.

math.CV↗
arXiv2012

Dynamics of multi-resonant biholomorphisms

The goal of this paper is to study the dynamics of holomorphic diffeomorphisms in C^n such that the resonances among the first 1<= r<= n eigenvalues of the differential are generated over N by a finite number of Q-linearly independent multi-indices (and more resonances are allowed for other eigenvalues). We give sharp conditions for the existence of basins of attraction where a Fatou coordinate can be defined. Furthermore, we obtain a generalization of the Leau-Fatou flower theorem, providing a complete description of the dynamics in a full neighborhood of the origin for 1-resonant parabolically attracting holomorphic germs in Poincare'-Dulac normal form.

math.CV↗