Fatou’s web
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Abstract:
Let $f$ be Fatou’s function, that is, $f(z)= z+1+e^{-z}$. We prove that the escaping set of $f$ has the structure of a ‘spider’s web’, and we show that this result implies that the non-escaping endpoints of the Julia set of $f$ together with infinity form a totally disconnected set. We also present a well-known transcendental entire function, due to Bergweiler, for which the escaping set is a spider’s web, and we point out that the same property holds for some families of functions.References
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Additional Information
- V. Evdoridou
- Affiliation: Department of Mathematics and Statistics, Walton Hall, The Open University, Milton Keynes MK7 6AA, United Kingdom
- Email: vasiliki.evdoridou@open.ac.uk
- Received by editor(s): October 26, 2015
- Received by editor(s) in revised form: February 4, 2016, and February 11, 2016
- Published electronically: June 3, 2016
- Communicated by: Jeremy Tyson
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 5227-5240
- MSC (2010): Primary 37F10; Secondary 30D05
- DOI: https://doi.org/10.1090/proc/13150
- MathSciNet review: 3556267