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Averages of holomorphic mappings and holomorphic retractions on convex hyperbolic domains

100%
Reich S., Shoikhet D.
Studia Mathematica
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1998
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tom 130
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nr 3
231-244
EN
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.
2
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A sharpened form of the inverse function theorem

100%
Elin M., Shoikhet D.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2019
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tom 73
|
nr 2
EN
In this note we establish an advanced version of the inverse function theorem and study some local geometrical properties like starlikeness and hyperbolic convexity of the inverse function under natural restrictions on the numerical range of the underlying mapping.
3
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Complex dynamical systems and the geometry of domains in Banach spaces

88%
Elin M., Reich S., Shoikhet D.
Instytut Matematyczny Polskiej Akademii Nauk
EN
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 equations. In addition, we introduce a new method involving admissible upper and lower bounds. Finally, we apply our dynamical approach to obtain several growth and covering theorems for star-like mappings on the open unit balls of Banach and Hilbert spaces.
4
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On the Rogosinski radius for holomorphic mappings and some of its applications

81%
Aizenberg L., Elin M., Shoikhet D.
Studia Mathematica
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2005
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tom 168
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nr 2
147-158
EN
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: $|∑_{n=0}^{∞} aₙzⁿ| < 1$, |z| < 1, then all its partial sums are less than 1 in the disk of radius 1/2: $|∑_{n=0}^{k} aₙzⁿ| < 1$, |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.
5
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Scientific life of Professor Yuri Kozitsky

81%
Holovatch Y., Prus S., Shoikhet D.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2019
|
tom 73
|
nr 2
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