Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 45, 2020, 511-531

SPIDERS' WEBS IN THE PUNCTURED PLANE

Vasiliki Evdoridou, David Martí-Pete and David J. Sixsmith

The Open University, School of Mathematics and Statistics
Milton Keynes MK7 6AA, United Kingdom; vasiliki.evdoridou 'at' open.ac.uk

Institute of Mathematics of the Polish Academy of Sciences
ul. Sniadeckich 8, 00-656 Warsaw, Poland; dmartipete 'at' impan.pl

University of Liverpool, Department of Mathematical Sciences
Liverpool L69 7ZL, United Kingdom; djs 'at' liverpool.ac.uk

Abstract. Many authors have studied sets, associated with the dynamics of a transcendental entire function, which have the topological property of being a spider's web. In this paper we adapt the definition of a spider's web to the punctured plane. We give several characterisations of this topological structure, and study the connection with the usual spider's web in C. We show that there are many transcendental self-maps of C^∗ for which the Julia set is such a spider's web, and we construct a transcendental self-map of C^∗ for which the escaping set I(f) has this structure and hence is connected. By way of contrast with transcendental entire functions, we conjecture that there is no transcendental self-map of C^∗ for which the fast escaping set A(f) is such a spider's web.

2010 Mathematics Subject Classification: Primary 37F10; Secondary 30D05.

Key words: Holomorphic dynamics, escaping set, punctured plane, spider's web.

Reference to this article: V. Evdoridou, D. Martí-Pete and D. J. Sixsmith: Spiders' webs in the punctured plane. Ann. Acad. Sci. Fenn. Math. 45 (2020), 511-531.

Full document as PDF file

https://doi.org/10.5186/aasfm.2020.4528