### Metric domains, holomorphic mappings and nonlinear semigroups.

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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.

Let $D$ be domain in a complex Banach space $X$, and let $\rho $ 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 $f:D\to X$ to be a generator of a $\rho $-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 $\rho $ the hyperbolic metric on $B$. We introduce the notion of a $\rho $-monotone mapping and obtain simple characterizations of generators...

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...

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: $|{\sum}_{n=0}^{\infty}a\u2099z\u207f|<1$, |z| < 1, then all its partial sums are less than 1 in the disk of radius 1/2: $|{\sum}_{n=0}^{k}a\u2099z\u207f|<1$, |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....

We present several characterizations and representations of semi-complete vector fields on the open unit balls in complex Euclidean and Hilbert spaces.

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