Annular itineraries for entire functions
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- by P. J. Rippon and G. M. Stallard PDF
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Abstract:
In order to analyse the way in which the size of the iterates of a transcendental entire function $f$ can behave, we introduce the concept of the annular itinerary of a point $z$. This is the sequence of non-negative integers $s_0s_1\ldots$ defined by \[ f^n(z)\in A_{s_n}(R),\;\;\text {for }n\ge 0, \] where $A_0(R)=\{z:|z|<R\}$ and \[ A_n(R)=\{z:M^{n-1}(R)\le |z|<M^n(R)\},\;\;n\ge 1. \] Here $M(r)$ is the maximum modulus of $f$ on $\{z:|z|=r\}$ and $R>0$ is so large that $M(r)>r$, for $r\ge R$.
We consider the different types of annular itineraries that can occur for any transcendental entire function $f$ and show that it is always possible to find points with various types of prescribed annular itineraries. The proofs use two new annuli covering results that are of wider interest.
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Additional Information
- P. J. Rippon
- Affiliation: Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom
- MR Author ID: 190595
- Email: p.j.rippon@open.ac.uk
- G. M. Stallard
- Affiliation: Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom
- MR Author ID: 292621
- Email: g.m.stallard@open.ac.uk
- Received by editor(s): January 24, 2013
- Published electronically: June 26, 2014
- Additional Notes: Both authors were supported by the EPSRC grant EP/H006591/1
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 367 (2015), 377-399
- MSC (2010): Primary 37F10; Secondary 30D05
- DOI: https://doi.org/10.1090/S0002-9947-2014-06354-X
- MathSciNet review: 3271265