Angular extents and trajectory slopes in the theory of holomorphic semigroups in the unit disk

We study relationships between the asymptotic behaviour of a non-elliptic semigroup of holomorphic self-maps of the unit disk and the geometry of its planar domain (the image of the Koenigs function). We establish a sufficient condition for the trajectories of the semigroup to converge to its Denjoy-Wolff point with a definite slope. We obtain as a corollary two previously known sufficient conditions

Centralizers of non-elliptic univalent self-maps and the embeddability problem in the unit disc

The embeddability problem is a very old and hard problem in discrete holomorphic iteration which deals with determining general conditions on a given univalent self-map $\varphi$ of the unit disc $\mathbb D$ in order to be contained in a continuous one-parameter semigroup. In this paper, we tackle this embedding problem by establishing different dichotomy results about the centralizer of $\varphi$ (i.e. the set of all univalent self-maps commuting with $\varphi$) which depend strongly on the dynamical character of $\varphi$. Our approach is, in part, based on a new technique to obtain simultaneous linearizations of two non-elliptic univalent self-maps of the unit disc, which might be interesting on their own. We also introduce and study several closed additive subsemigroups of the complex plane that collect the main features of the centralizer of $\varphi$ and which play a prominent position in those dichotomy results.Comment: Ver3: The most important changes concern our viewpoint on the result about the simultaneous linearization of commuting self-maps, Theorem 4.1. Accordingly, a part of Introduction and Section 4 have been re-organized. Ver2: an error in the list of references and a misprint in the Introduction are corrected; references [7,23] are adde

Innovative development of Russian coastal regions: North–south divergence

Coastal regions are territorial social systems whose socio-economic and innovative development is strongly influenced by the factor of coastalisation. The effect of movement to the sea determines the dynamics of settlement systems as well as their economic and infrastructure development. This holds for transport, logistics, information and communications, industrial, and other infrastructure. Coastal regions are so diverse that it is impossible to construct a development model that will fit all of them. One can speak only of general trends. This study focuses on identifying differences between the innovation systems of northern and southern coastal regions within the same country. The geographical scope of the study is four Russian coastal territories: Murmansk and Arkhangelsk in the Baltic Sea region and Rostov and Krasnodar in the Azov-Black Sea region. Methodologically, this study carries out a comparative assessment of heterogeneity of innovative development at municipal and interregional levels, using four groups of indicators: human capital, economic growth and clustering, innovation and digitalisation, and quality and standards of living. All these components are vital for regional innovative development. A statistical assessment is supplemented by a qualitative analysis of spatial patterns of innovation capital accumulation; the agglomeration factor is taken into account. It is shown that northern and southern coastal regions perform very differently on innovative development, the latter doing better than the former. Three main models of innovation generation, implementation, and accumulation of coastal regions are described. Each is associated with a different way to benefit from proximity to the sea. These are maritime activities, maritime transport, and the economic use of recreational, natural and climatic resources