Filippo Bracci

Pluripotential theory, semigroups and boundary behavior of infinitesimal generators in strongly convex domains

We characterize infinitesimal generators of semigroups of holomorphic self-maps of strongly convex domains using the pluricomplex Green function and the pluricomplex Poisson kernel. Moreover, we study boundary regular fixed points of semigroups. Among other things, we characterize boundary regular fixed points both in terms of the boundary behavior of infinitesimal generators and in terms of pluripotential theory.

Identity principles for commuting holomorphic self-maps of the unit disc

Abstract Let f,g be two commuting holomorphic self-maps of the unit disc. If f and g agree at the common Wolff point up to a certain order of derivatives (no more than 3 if the Wolff point is on the unit circle), then f≡g.

The pluricomplex Poisson kernel for strongly convex domains

Let D be a bounded strongly convex domain in the complex space of dimension n. For a fixed point p epsilon partial derivative D, we consider the solution of a homogeneous complex Monge-Ampere equation with a simple pole at p. We prove that such a solution enjoys many properties of the classical Poisson kernel in the unit disc and thus deserves to be called the pluricomplex Poisson kernel of D with pole at p. In particular we discuss extremality properties (such as a generalization of the classical Phragmen-Lindelof theorem), relations with the pluricomplex Green function of D, uniqueness in terms of the associated foliation and boundary behaviors. Finally, using such a kernel we obtain explicit reproducing formulas for plurisubharmonic functions.

Dynamics of one-resonant biholomorphisms

Our first main result is a construction of a simple formal normal form for holomorphic diffeomorphisms in C^n whose differentials have one-dimensional family of resonances in the first m eigenvalues, m <= n (but more resonances are allowed for other eigenvalues). Next, we provide invariants and give conditions for the existence of basins of attraction. Finally, we give applications and examples demonstrating the sharpness of our conditions.

Valiron's construction in higher dimension

We consider holomorphic self-maps ϕ of the unit ball B N in C N (N =1 , 2, 3 ,... ). In the one-dimensional case, when ϕ has no fixed points in D := B 1 and is of hyperbolic type, there is a classical renormalization procedure due to Valiron which allows to semi-linearize the map ϕ, and therefore, in this case, the dynamical properties of ϕ are well understood. In what follows, we generalize the classical Valiron construction to higher dimensions under some weak assumptions on ϕ at its Denjoy-Wolff point. As a result, we construct a semi-conjugation σ, which maps the ball into the right half-plane of C ,a nd solves the functional equation σ ◦ ϕ = λσ ,w here λ> 1i s the (inverse of the) boundary dilation coefficient at the Denjoy-Wolff point of ϕ.

HYPERBOLICITY IN UNBOUNDED CONVEX DOMAINS

Abstract We provide several equivalent characterizations of Kobayashi hyperbolicity in unbounded convex domains in terms of peak and anti-peak functions at infinity, affine lines, Bergman metric and iteration theory.

RESIDUES FOR SINGULAR PAIRS AND DYNAMICS OF BIHOLOMORPHIC MAPS OF SINGULAR SURFACES

We prove the existence of a parabolic curve for a germ of biholomorphic map tangent to the identity at an isolated singular point of a surface under some conditions. For this purpose, we present a Camacho–Sad type index theorem for fixed curves of biholomorphic maps of singular surfaces and develop a local intersection theory of curves in singular surfaces from an analytic approach by means of Grothendieck residues.

DILATATION AND ORDER OF CONTACT FOR HOLOMORPHIC SELF-MAPS OF STRONGLY CONVEX DOMAINS

Let

Classical and Stochastic Löwner–Kufarev Equations

D

Embedding univalent functions in filtering Loewner chains in higher dimension

be a bounded strongly convex domain and let