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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: 75 (2 UL)Limit functions of discrete dynamical systems ; Meyrath, Thierry ; in Conformal Geometry & 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: 60 (3 UL)Universality properties of Taylor series inside the domain of holomorphy ; Meyrath, Thierry ; in Journal of Mathematical Analysis & 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: 115 (4 UL) |
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