## Cantor bouquets in spiders’ webs

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- by Yannis Dourekas PDF
- Conform. Geom. Dyn.
**24**(2020), 1-28 Request permission

## Abstract:

For many transcendental entire functions, the escaping set has the structure of a Cantor bouquet, consisting of uncountably many disjoint curves. Rippon and Stallard showed that there are many functions for which the escaping set has a new connected structure known as an infinite spider’s web. We investigate a connection between these two topological structures for a certain class of sums of exponentials.## References

- Jan M. Aarts and Lex G. Oversteegen,
*The geometry of Julia sets*, Trans. Amer. Math. Soc.**338**(1993), no. 2, 897–918. MR**1182980**, DOI 10.1090/S0002-9947-1993-1182980-3 - I. N. Baker,
*Repulsive fixpoints of entire functions*, Math. Z.**104**(1968), 252–256. MR**226009**, DOI 10.1007/BF01110294 - I. N. Baker,
*Wandering domains in the iteration of entire functions*, Proc. London Math. Soc. (3)**49**(1984), no. 3, 563–576. MR**759304**, DOI 10.1112/plms/s3-49.3.563 - Krzysztof Barański, Xavier Jarque, and Lasse Rempe,
*Brushing the hairs of transcendental entire functions*, Topology Appl.**159**(2012), no. 8, 2102–2114. MR**2902745**, DOI 10.1016/j.topol.2012.02.004 - Walter Bergweiler and Bogusława Karpińska,
*On the Hausdorff dimension of the Julia set of a regularly growing entire function*, Math. Proc. Cambridge Philos. Soc.**148**(2010), no. 3, 531–551. MR**2609307**, DOI 10.1017/S0305004109990491 - Paul Blanchard,
*Complex analytic dynamics on the Riemann sphere*, Bull. Amer. Math. Soc. (N.S.)**11**(1984), no. 1, 85–141. MR**741725**, DOI 10.1090/S0273-0979-1984-15240-6 - Clara Bodelón, Robert L. Devaney, Michael Hayes, Gareth Roberts, Lisa R. Goldberg, and John H. Hubbard,
*Hairs for the complex exponential family*, Internat. J. Bifur. Chaos Appl. Sci. Engrg.**9**(1999), no. 8, 1517–1534. MR**1721835**, DOI 10.1142/S0218127499001061 - Michael Brin and Garrett Stuck,
*Introduction to dynamical systems*, Cambridge University Press, Cambridge, 2002. MR**1963683**, DOI 10.1017/CBO9780511755316 - Robert L. Devaney and Folkert Tangerman,
*Dynamics of entire functions near the essential singularity*, Ergodic Theory Dynam. Systems**6**(1986), no. 4, 489–503. MR**873428**, DOI 10.1017/S0143385700003655 - Robert L. Devaney and MichałKrych,
*Dynamics of $\textrm {exp}(z)$*, Ergodic Theory Dynam. Systems**4**(1984), no. 1, 35–52. MR**758892**, DOI 10.1017/S014338570000225X - John Milnor,
*Dynamics in one complex variable*, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR**2193309** - J. W. Osborne,
*Spiders’ webs and locally connected Julia sets of transcendental entire functions*, Ergodic Theory Dynam. Systems**33**(2013), no. 4, 1146–1161. MR**3082543**, DOI 10.1017/S0143385712000259 - G. Pólya and G. Szegő,
*Problems and theorems in analysis. Vol. II*, Revised and enlarged translation by C. E. Billigheimer of the fourth German edition, Springer Study Edition, Springer-Verlag, New York-Heidelberg, 1976. Theory of functions, zeros, polynomials, determinants, number theory, geometry. MR**0465631**, DOI 10.1007/978-1-4757-6292-1 - P. J. Rippon and G. M. Stallard,
*Fast escaping points of entire functions*, Proc. Lond. Math. Soc. (3)**105**(2012), no. 4, 787–820. MR**2989804**, DOI 10.1112/plms/pds001 - P. J. Rippon and G. M. Stallard,
*On sets where iterates of a meromorphic function zip towards infinity*, Bull. London Math. Soc.**32**(2000), no. 5, 528–536. MR**1767705**, DOI 10.1112/S002460930000730X - Dierk Schleicher and Johannes Zimmer,
*Escaping points of exponential maps*, J. London Math. Soc. (2)**67**(2003), no. 2, 380–400. MR**1956142**, DOI 10.1112/S0024610702003897 - David J. Sixsmith,
*Julia and escaping set spiders’ webs of positive area*, Int. Math. Res. Not. IMRN**19**(2015), 9751–9774. MR**3431610**, DOI 10.1093/imrn/rnu245 - E. C. Titchmarsh,
*The theory of functions*, 2nd ed., Oxford University Press, Oxford, 1939. MR**3728294**

## Additional Information

**Yannis Dourekas**- Affiliation: School of Mathematics and Statistics, Open University, Milton Keynes MK7 6AA, United Kingdom
- Email: ioandour@gmail.com
- Received by editor(s): September 20, 2019
- Published electronically: January 7, 2020
- © Copyright 2020 American Mathematical Society
- Journal: Conform. Geom. Dyn.
**24**(2020), 1-28 - MSC (2010): Primary 37F10
- DOI: https://doi.org/10.1090/ecgd/346
- MathSciNet review: 4047936