Eremenko points and the structure of the escaping set
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- by P. J. Rippon and G. M. Stallard PDF
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Abstract:
Much recent work on the iterates of a transcendental entire function $f$ has been motivated by Eremenko’s conjecture that all the components of the escaping set $I(f)$ are unbounded. We prove several general results about the topological structure of $I(f)$ including the fact that if $I(f)$ is disconnected, then it contains uncountably many pairwise disjoint unbounded continua, all of which are subsets of the fast escaping set. We give analogous results for the intersection of $I(f)$ with the Julia set when multiply connected wandering domains are not present, and show that completely different results hold when such wandering domains are present. In proving these, we obtain the unexpected result that some types of multiply connected wandering domains have complementary components with no interior, indeed uncountably many.References
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Additional Information
- P. J. Rippon
- Affiliation: School of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom
- MR Author ID: 190595
- Email: phil.rippon@open.ac.uk
- G. M. Stallard
- Affiliation: School of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom
- MR Author ID: 292621
- Email: gwyneth.stallard@open.ac.uk
- Received by editor(s): September 15, 2017
- Received by editor(s) in revised form: March 12, 2018
- Published electronically: June 10, 2019
- Additional Notes: Both authors were supported by EPSRC grant EP/K031163/1.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 3083-3111
- MSC (2010): Primary 30D05, 37F10
- DOI: https://doi.org/10.1090/tran/7673
- MathSciNet review: 3988603