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Averages of holomorphic mappings and holomorphic retractions on convex hyperbolic domains

Simeon Reich;  David Shoikhet — 1998

Studia Mathematica

Let D be a hyperbolic convex domain in a complex Banach space. Let the mapping F ∈ Hol(D,D) be bounded on each subset strictly inside D, and have a nonempty fixed point set ℱ in D. We consider several methods for constructing retractions onto ℱ under local assumptions of ergodic type. Furthermore, we study the asymptotic behavior of the Cesàro averages of one-parameter semigroups generated by holomorphic mappings.

Semigroups and generators on convex domains with the hyperbolic metric

Simeon Reich;  David Shoikhet — 1997

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let D be domain in a complex Banach space X , and let ρ be a pseudometric assigned to D by a Schwarz-Pick system. In the first section of the paper we establish several criteria for a mapping f : D → X to be a generator of a ρ -nonexpansive semigroup on D in terms of its nonlinear resolvent. In the second section we let X = H be a complex Hilbert space, D = B the open unit ball of H , and ρ the hyperbolic metric on B . We introduce the notion of a ρ -monotone mapping and obtain simple characterizations of generators...

Complex dynamical systems and the geometry of domains in Banach spaces

Mark Elin;  Simeon Reich;  David Shoikhet — 2004

We consider semigroups of holomorphic self-mappings on domains in Hilbert and Banach spaces, and then develop a new dynamical approach to the study of geometric properties of biholomorphic mappings. We establish, for example, several flow invariance conditions and find parametric representations of semicomplete vector fields. In order to examine the asymptotic behavior of these semigroups, we use diverse tools such as hyperbolic metric theory and estimates of solutions of generalized differential...

On the Rogosinski radius for holomorphic mappings and some of its applications

Lev Aizenberg;  Mark Elin;  David Shoikhet — 2005

Studia Mathematica

The well known theorem of Rogosinski asserts that if the modulus of the sum of a power series is less than 1 in the open unit disk: | ∑ n = 0 ∞ a ₙ z ⁿ | < 1 , |z| < 1, then all its partial sums are less than 1 in the disk of radius 1/2: | ∑ n = 0 k a ₙ z ⁿ | < 1 , |z| < 1/2, and this radius is sharp. We present a generalization of this theorem to holomorphic mappings of the open unit ball into an arbitrary convex domain. Other multidimensional analogs of Rogosinski’s theorem as well as some applications to dynamical systems are considered....

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