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Non-normality, topological transitivity and expanding families Meyrath, Thierry ; in Mathematical Proceedings of the Cambridge Philosophical Society (2022), 173(3), 511-523 We investigate the behaviour of families of meromorphic functions in the neighbourhood of points of non-normality and prove certain covering properties that complement Montel’s Theorem. In particular, we ... [more ▼] We investigate the behaviour of families of meromorphic functions in the neighbourhood of points of non-normality and prove certain covering properties that complement Montel’s Theorem. In particular, we also obtain characterisations of non-normality in terms of such properties. [less ▲] Detailed reference viewed: 54 (2 UL)Mixing Taylor shifts and universal Taylor series ; Meyrath, Thierry ; in Bulletin of the London Mathematical Society (2015), 47 It is known that, generically, Taylor series of functions holomorphic in a simply connected complex domain exhibit maximal erratic behaviour outside the domain (so-called universality) and overconvergence ... [more ▼] It is known that, generically, Taylor series of functions holomorphic in a simply connected complex domain exhibit maximal erratic behaviour outside the domain (so-called universality) and overconvergence in parts of the domain. Our aim is to show how the theory of universal Taylor series can be put into the framework of linear dynamics. This leads to a unified approach to universality and overconvergence and yields new insight into the boundary behaviour of Taylor series. [less ▲] Detailed reference viewed: 105 (2 UL)Limit functions of discrete dynamical systems ; Meyrath, Thierry ; in Conformal Geometry and Dynamics (2014), 18 In the theory of dynamical systems, the notion of ω-limit sets of points is classical. In this paper, the existence of limit functions on subsets of the underlying space is treated. It is shown that in ... [more ▼] In the theory of dynamical systems, the notion of ω-limit sets of points is classical. In this paper, the existence of limit functions on subsets of the underlying space is treated. It is shown that in the case of topologically mixing systems on appropriate metric spaces (X, d), the existence of at least one limit function on a compact subset A of X implies the existence of plenty of them on many supersets of A. On the other hand, such sets necessarily have to be small in various respects. The results for general discrete systems are applied in the case of Julia sets of rational functions and in particular in the case of the existence of Siegel disks. [less ▲] Detailed reference viewed: 109 (4 UL)On the behaviour of the successive derivatives of meromorphic functions on the final set Meyrath, Thierry ; in Journal d'analyse mathématique (2013), 120(1), 131-149 We study the behaviour of the sequence of successive derivatives of meromorphic functions at points of the so-called final set. We prove that, whereas in many cases this sequence tends to ∞, for a special ... [more ▼] We study the behaviour of the sequence of successive derivatives of meromorphic functions at points of the so-called final set. We prove that, whereas in many cases this sequence tends to ∞, for a special class of meromorphic functions, it may have extremely wild behaviour. We also prove a connection between the derivatives of meromorphic functions from this class and so-called Dirichlet sets. [less ▲] Detailed reference viewed: 111 (0 UL)Universality properties of Taylor series inside the domain of holomorphy ; Meyrath, Thierry ; in Journal of Mathematical Analysis and Applications (2011), 383(1), 234-238 It is proven that the Taylor series of functions holomorphic in $\C_{\infty} \setminus \{1\}$ generically have certain universality properties on small sets outside the unit disk. Moreover, it is shown ... [more ▼] It is proven that the Taylor series of functions holomorphic in $\C_{\infty} \setminus \{1\}$ generically have certain universality properties on small sets outside the unit disk. Moreover, it is shown that such sets necessarily are polar sets. [less ▲] Detailed reference viewed: 152 (5 UL) |
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