On the Hausdorff dimension of the Julia set of a regularly growing entire function

We show that if the growth of a transcendental entire function f is sufficiently regular, then the Julia set and the escaping set of f have Hausdorff dimension 2.Comment: 21 page

Absorbing sets and Baker domains for holomorphic maps

We consider holomorphic maps $f: U \to U$ for a hyperbolic domain $U$ in the complex plane, such that the iterates of $f$ converge to a boundary point $\zeta$ of $U$. By a previous result of the authors, for such maps there exist nice absorbing domains $W \subset U$. In this paper we show that $W$ can be chosen to be simply connected, if $f$ has parabolic I type in the sense of the Baker--Pommerenke--Cowen classification of its lift by a universal covering (and $\zeta$ is not an isolated boundary point of $U$). Moreover, we provide counterexamples for other types of the map $f$ and give an exact characterization of parabolic I type in terms of the dynamical behaviour of $f$

Brushing the hairs of transcendental entire functions

Let f be a hyperbolic transcendental entire function of finite order in the Eremenko-Lyubich class (or a finite composition of such maps), and suppose that f has a unique Fatou component. We show that the Julia set of $f$ is a Cantor bouquet; i.e. is ambiently homeomorphic to a straight brush in the sense of Aarts and Oversteegen. In particular, we show that any two such Julia sets are ambiently homeomorphic. We also show that if f\in\B has finite order (or is a finite composition of such maps), but is not necessarily hyperbolic, then the Julia set of f contains a Cantor bouquet. As part of our proof, we describe, for an arbitrary function f\in\B, a natural compactification of the dynamical plane by adding a "circle of addresses" at infinity.Comment: 19 pages. V2: Small number of minor corrections made from V

Initial remarks on artificial intelligence and axiological foundations of labour law

Progressive technological development forces a new look at the basic constructs of labour law. The science of labour law has so far paid little attention to the legal analysis of the risks associated with the ever-increasing interaction between people and technological tools, both in the form of advanced machinery as well as software used to manage enterprises and production processes. At the same time, questions about the future of labour law have long been posed in the science of labour law, primarily raising the need to realize the principle of social justice when work is not performed within the framework of the traditionally understood employment relationship (under employee subordination). The present study is a  voice in this discussion, focusing on the issue of artificial intelligence in relation to the axiological foundations of labour law. The article analyzes the humanisation of work and human interaction in the world of new technologies primarily from the perspective of the central value of labour law, which is the inherent and inalienable dignity of the employee as a human being. There is also a consideration of the concept of autonomisation of the employee in the work process using artificial intelligence