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Santiago Diaz-Madrigal

Publications and source records attributed to Santiago Diaz-Madrigal.

15 recordsLinked to original sources
arXiv2016

Topological invariants for semigroups of holomorphic self-maps of the unit disc

Let $(φ_t)$, $(ϕ_t)$ be two one-parameter semigroups of holomorphic self-maps of the unit disc $\mathbb D\subset \mathbb C$. Let $f:\mathbb D \to \mathbb D$ be a homeomorphism. We prove that, if $f \circ ϕ_t=φ_t \circ f$ for all $t\geq 0$, then $f$ extends to a homeomorphism of $\bar{\mathbb D}$ outside exceptional maximal contact arcs (in particular, for elliptic semigroups, $f$ extends to a homeomorphism of $\bar{\mathbb D}$). Using this result, we study topological invariants for one-parameter semigroups of holomorphic self-maps of the unit disc.

math.CV↗
arXiv2014

Compact and weakly compact composition operators from the Bloch space into Möbius invariant spaces

We obtain exhaustive results and treat in a unified way the question of boundedness, compactness, and weak compactness of composition operators from the Bloch space into any space from a large family of conformally invariant spaces that includes the classical spaces like $BMOA$, $Q_α$, and analytic Besov spaces $B^p$. In particular, by combining techniques from both complex and functional analysis, we prove that in this setting weak compactness is equivalent to compactness. For the operators into the corresponding "small" spaces we also characterize the boundedness and show that it is equivalent to compactness.

math.FA↗
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

Local duality in Loewner equations

Among diversity of frameworks and constructions introduced in Loewner Theory by different authors, one can distinguish two closely related but still different ways of reasoning, which colloquially may be described as "increasing" and "decreasing". In this paper we review in short the main types of (deterministic) Loewner evolution discussed in the literature and describe in detail the local duality between "increasing" and "decreasing" cases within the general unifying approach in Loewner Theory proposed recently in [Bracci et al. to appear in J Reine Angew Math; arXiv:0807.1594v1], [Bracci et al. in Math Ann 344:947-962, 2009; arXiv:0807.1715v1], [Contreras et al. in Revista Matemática Iberoamericana 26:975-1012, 2010; arXiv:0902.3116v1]. In particular, we extend several results of R.O.Bauer [in J Math Anal Appl 302:484-501, 2005; arXiv:math/0306130v1], which deal with the chordal Loewner evolution, to this general setting. Although the duality is given by a simple change of the parameter, not all the results for the "decreasing" case can be obtained by mere translating the corresponding results for the "increasing" case. In particular, as a byproduct of establishing local duality between evolution families and their "decreasing" counterparts we obtain a new characterization of generalized Loewner chains.

math.CV↗
arXiv2011

Loewner Theory in annulus II: Loewner chains

Loewner Theory, based on dynamical viewpoint, proved itself to be a powerful tool in Complex Analysis and its applications. Recently Bracci et al [Bracci et al, to appear in J. Reine Angew. Math. Available on ArXiv 0807.1594; Bracci et al, Math. Ann. 344(2009), 947--962; Contreras et al, Revista Matematica Iberoamericana 26(2010), 975--1012] have proposed a new approach bringing together all the variants of the (deterministic) Loewner Evolution in a simply connected reference domain. This paper is devoted to the construction of a general version of Loewner Theory for the annulus launched in [ArXiv 1011.4253]. We introduce the general notion of a Loewner chain over a system of annuli and obtain a 1-to-1 correspondence between Loewner chains and evolution families in the doubly connected setting similar to that in the Loewner Theory for the unit disk. Futhermore, we establish a conformal classification of Loewner chains via the corresponding evolution families and via semicomplete weak holomorphic vector fields. Finally, we extend the explicit characterization of the semicomplete weak holomorphic vector fields obtained in [ArXiv 1011.4253] to the general case.

math.CV↗
arXiv2010

Loewner Theory in annulus I: evolution families and differential equations

Loewner Theory, based on dynamical viewpoint, is a powerful tool in Complex Analysis, which plays a crucial role in such important achievements as the proof of famous Bieberbach's conjecture and well-celebrated Schramm's Stochastic Loewner Evolution (SLE). Recently Bracci et al [Bracci et al, to appear in J. Reine Angew. Math. Available on ArXiv 0807.1594; Bracci et al, Math. Ann. 344(2009), 947--962; Contreras et al, Revista Matematica Iberoamericana 26(2010), 975--1012] have proposed a new approach bringing together all the variants of the (deterministic) Loewner Evolution in a simply connected reference domain. We construct an analogue of this theory for the annulus. In this paper, the first of two articles, we introduce a general notion of an evolution family over a system of annuli and prove that there is a 1-to-1 correspondence between such families and semicomplete weak holomorphic vector fields. Moreover, in the non-degenerate case, we establish a constructive characterization of these vector fields analogous to the non-autonomous Berkson - Porta representation of Herglotz vector fields in the unit disk [Bracci et al, to appear in J. Reine Angew. Math. Available on ArXiv 0807.1594].

math.CV↗
arXiv2010

Geometry behind chordal Loewner chains

Loewner Theory is a deep technique in Complex Analysis affording a basis for many further important developments such as the proof of famous Bieberbach's conjecture and well-celebrated Schramm's Stochastic Loewner Evolution (SLE). It provides analytic description of expanding domains dynamics in the plane. Two cases have been developed in the classical theory, namely the {\it radial} and the {\it chordal} Loewner evolutions, referring to the associated families of holomorphic self-mappings being normalized at an internal or boundary point of the reference domain, respectively. Recently there has been introduced a new approach [arXiv:0807.1594v1, arXiv:0807.1715v1, arXiv:0902.3116v1] bringing together, and containing as quite special cases, radial and chordal variants of Loewner Theory. In the framework of this approach we address the question what kind of systems of simply connected domains can be described by means of Loewner chains of chordal type. As an answer to this question we establish a necessary and sufficient condition for a set of simply connected domains to be the range of a generalized Loewner chain of chordal type. We also provide an easy-to-check geometric sufficient condition for that. In addition, we obtain analogous results for the less general case of chordal Loewner evolution considered by I.A.Aleksandrov, S.T.Aleksandrov and V.V.Sobolev, 1983; V.V.Goryainov and I.Ba, 1992; R.O.Bauer, 2005 [arXiv:math/0306130v1].

math.CV↗
arXiv2009

A Non-Autonomous Version Of The Denjoy-Wolff Theorem

The aim of this work is to establish the celebrated Denjoy-Wolff Theorem in the context of generalized Loewner chains. In contrast with the classical situation where essentially convergence to a certain point in the closed unit disk is the unique possibility, several new dynamical phenomena appear in this framework. Indeed, $ω$-limits formed by suitable closed arcs of circumferences appear now as natural possibilities of asymptotic dynamical behavior.

math.DS↗
arXiv2009

Semigroups versus evolution families in the Loewner theory

We show that an evolution family of the unit disc is commuting if and only if the associated Herglotz vector field has separated variables. This is the case if and only if the evolution family comes from a semigroup of holomorphic self-maps of the disc.

math.CV↗
arXiv2009

Loewner chains in the unit disk

In this paper we introduce a general version of the notion of Loewner chains which comes from the new and unified treatment, given in [arXiv:0807.1594], of the radial and chordal variant of the Loewner differential equation, which is of special interest in geometric function theory as well as for various developments it has given rise to, including the famous Schramm-Loewner evolution. In this very general setting, we establish a deep correspondence between these chains and the evolution families introduced in [arXiv:0807.1594]. Among other things, we show that, up to a Riemann map, such a correspondence is one-to-one. In a similar way as in the classical Loewner theory, we also prove that these chains are solutions of a certain partial differential equation which resembles (and includes as a very particular case) the classical Loewner-Kufarev PDE.

math.CV↗
arXiv2008

Evolution Families and the Loewner Equation I: The Unit Disc

In this paper we introduce a general version of the Loewner differential equation which allows us to present a new and unified treatment of both the radial equation introduced in 1923 by K. Loewner and the chordal equation introduced in 2000 by O. Schramm. In particular, we prove that evolution families in the unit disc are in one to one correspondence with solutions to this new type of Loewner equations. Also, we give a Berkson-Porta type formula for non-autonomous weak holomorphic vector fields which generate such Loewner differential equations and study in detail geometric and dynamical properties of evolution families.

math.CV↗
arXiv2007

Aleksandrov-Clark measures and semigroups of analytic functions in the unit disc

In this paper we prove a formula describing the infinitesimal generator of a continuous semigroup $(\v_t)$ of holomorphic self-maps of the unit disc with respect to a boundary regular fixed point. The result is based on Alexandrov-Clark measures techniques. In particular we prove that the Alexandrov-Clark measure of $(\v_t)$ at a boundary regular fixed points is differentiable (in the weak$^\ast$-topology) with respect to $t$.

math.CV↗
arXiv2006

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.

math.CV↗
arXiv2006

Infinitesimal generators associated with semigroups of linear fractional maps

We characterize the infinitesimal generator of a semigroup of linear fractional self-maps of the unit ball in $\mathbb C^n$, $n\geq 1$. For the case $n=1$ we also completely describe the associated Koenigs function and we solve the embedding problem from a dynamical point of view, proving, among other things, that a generic semigroup of holomorphic self-maps of the unit disc is a semigroup of linear fractional maps if and only if it contains a linear fractional map for some positive time.

math.CV↗