Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-11T05:11:17.185Z Has data issue: false hasContentIssue false
  • English
  • Français

On slow escaping and non-escaping points of quasimeromorphic mappings

Part of: Geometric function theory Entire and meromorphic functions, and related topics Complex dynamical systems

Published online by Cambridge University Press:  22 January 2020

LUKE WARREN Open the ORCID record for LUKE WARREN [Opens in a new window]
LUKE WARREN*
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK email luke.warren@nottingham.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that for any quasimeromorphic mapping with an essential singularity at infinity, there exist points whose iterates tend to infinity arbitrarily slowly. This extends a result by Nicks for quasiregular mappings, and Rippon and Stallard for transcendental meromorphic functions on the complex plane. We further establish a new result for the growth rate of quasiregular mappings near an essential singularity, and briefly extend some results regarding the bounded orbit set and the bungee set to the quasimeromorphic setting.

Keywords

quasiregular mappingsquasimeromorphic mappingsescaping setbungee setslow escape

MSC classification

Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions
Secondary: 30C65: Quasiconformal mappings in $^n$, other generalizations 30D05: Functional equations in the complex domain, iteration and composition of analytic functions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 4 , April 2021 , pp. 1190 - 1216
Check for updates
Copyright
© Cambridge University Press, 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bergweiler, W.. Fixed points of composite entire and quasiregular maps. Ann. Acad. Sci. Fenn. Math. 31 (2006), 523–540.Google Scholar
Bergweiler, W.. On the set where the iterates of an entire function are bounded. Proc. Amer. Math. Soc. 140(3) (2012), 847–853.10.1090/S0002-9939-2011-11456-4CrossRefGoogle Scholar
Bergweiler, W.. Fatou–Julia theory for non-uniformly quasiregular maps. Ergod. Th. & Dynam. Sys. 33(1) (2013), 1–23.10.1017/S0143385711000915CrossRefGoogle Scholar
Bergweiler, W., Drasin, D. and Fletcher, A. N.. The fast escaping set of a quasiregular mapping. Anal. Math. Phys. 4 (2014), 83–98.10.1007/s13324-014-0078-9CrossRefGoogle Scholar
Bergweiler, W., Fletcher, A. N., Langley, J. and Mayer, V.. The escaping set of a quasiregular mapping. Proc. Amer. Math. Soc. 137 (2009), 641–651.10.1090/S0002-9939-08-09609-3CrossRefGoogle Scholar
Bergweiler, W. and Hinkkanen, A.. On semiconjugation of entire functions. Math. Proc. Cambridge Philos. Soc. 126 (1999), 565–574.10.1017/S0305004198003387CrossRefGoogle Scholar
Bergweiler, W. and Nicks, D. A.. Foundations for an iteration theory of entire quasiregular maps. Israel J. Math. 201(1) (2014), 147–184.10.1007/s11856-014-1081-4CrossRefGoogle Scholar
Bergweiler, W. and Peter, J.. Escape rate and Hausdorff measure for entire functions. Math. Z. 274 (2013), 551–572.10.1007/s00209-012-1085-xCrossRefGoogle Scholar
Callender, E. D.. Hölder continuity of n-dimensional quasiconformal mappings. Pacific J. Math. 10(2) (1960), 499–515.10.2140/pjm.1960.10.499CrossRefGoogle Scholar
Domínguez, P.. Dynamics of transcendental meromorphic functions. Ann. Acad. Sci. Fenn. Math. 23 (1998), 225–250.Google Scholar
Eremenko, A. E.. On the iteration of entire functions. Dynamical Systems and Ergodic Theory (Warsaw 1986) (Banach Center Publications, 23) . PWN, Warsaw, 1989, pp. 339–345.Google Scholar
Eremenko, A. E. and Ljubich, M. Ju.. Examples of entire functions with pathological dynamics. J. Lond. Math. Soc. 36(2) (1987), 458–468.10.1112/jlms/s2-36.3.458CrossRefGoogle Scholar
Martí-Pete, D.. The escaping set of transcendental self-maps of the punctured plane. Ergod. Th. & Dynam. Sys. 38 (2018), 739–760.10.1017/etds.2016.36CrossRefGoogle Scholar
Martio, O., Rickman, S. and Väisälä, J.. Definitions for quasiregular mappings. Ann. Acad. Sci. Fenn. Ser. A I Math. 448 (1969), 1–40.Google Scholar
Martio, O., Rickman, S. and Väisälä, J.. Distortion and singularities of quasiregular mappings. Ann. Acad. Sci. Fenn. Ser. A I Math. 465 (1970), 1–13.Google Scholar
Miniowitz, R.. Normal families of quasimeromorphic mappings. Proc. Amer. Math. Soc. 84(1) (1982), 35–43.10.1090/S0002-9939-1982-0633273-XCrossRefGoogle Scholar
Nicks, D. A.. Slow escaping points of quasiregular mappings. Math. Z. 284 (2016), 1053–1071.10.1007/s00209-016-1687-9CrossRefGoogle Scholar
Nicks, D. A. and Sixsmith, D. J.. The dynamics of quasiregular maps of punctured space. Indiana Univ. Math. J. 68(1) (2019), 323–352.10.1512/iumj.2019.68.7556CrossRefGoogle Scholar
Nicks, D. A. and Sixsmith, D. J.. The bungee set in quasiregular dynamics. Bull. Lond. Math. Soc. 51(1) (2019), 120–128.10.1112/blms.12215CrossRefGoogle Scholar
Osborne, J. W.. Connectedness properties of the set where the iterates of an entire function are bounded. Math. Proc. Cambridge Philos. Soc. 155(3) (2013), 391–410.10.1017/S0305004113000455CrossRefGoogle Scholar
Osborne, J. W. and Sixsmith, D. J.. On the set where the iterates of an entire function are neither escaping nor bounded. Ann. Acad. Sci. Fenn. Math. 41(2) (2016), 561–578.10.5186/aasfm.2016.4134CrossRefGoogle Scholar
Radó, T. and Reichelderfer, P. V.. Continuous Transformations in Analysis (Grundlehren der math. Wissenschaften, 75) . Springer, Berlin, 1955.10.1007/978-3-642-85989-2CrossRefGoogle Scholar
Reshetnyak, Y.. Space mappings with bounded distortion. Sibirsk. Mat. Zh. 8 (1967), 629–659.Google Scholar
Reshetnyak, Y.. On the condition of the boundedness of index for mappings with bounded distortion. Sibirsk. Mat. Zh. 9 (1968), 368–374.Google Scholar
Rickman, S.. On the number of omitted values of entire quasiregular mappings. J. Analyse Math. 37 (1980), 100–117.10.1007/BF02797681CrossRefGoogle Scholar
Rickman, S.. The analogue of Picard’s theorem for quasiregular mappings in dimension three. Acta Math. 154 (1985), 195–242.10.1007/BF02392472CrossRefGoogle Scholar
Rickman, S.. Quasiregular mappings (Ergebnisse der Mathematik und ihrer Grenzgebiete, 26) . Springer, Berlin, 1993.10.1007/978-3-642-78201-5CrossRefGoogle Scholar
Rippon, P. J. and Stallard, G. M.. On sets where iterates of a meromorphic function zip towards infinity. Bull. London Math. Soc. 32 (2000), 528–536.10.1112/S002460930000730XCrossRefGoogle Scholar
Rippon, P. J. and Stallard, G. M.. Escaping points of meromorphic functions with a finite number of poles. J. Anal. Math. 96 (2005), 225–245.10.1007/BF02787829CrossRefGoogle Scholar
Rippon, P. J. and Stallard, G. M.. Escaping points of entire functions of small growth. Math. Z. 261(3) (2009), 557–570.10.1007/s00209-008-0339-0CrossRefGoogle Scholar
Rippon, P. J. and Stallard, G. M.. Slow escaping points of meromorphic functions. Trans. Amer. Math. Soc. 363(8) (2011), 4171–4201.10.1090/S0002-9947-2011-05158-5CrossRefGoogle Scholar
Rottenfusser, G., Rückert, J., Rempe, L. and Schleicher, D.. Dynamic rays of bounded-type entire functions. Ann. Math. 173(1) (2011), 77–125.10.4007/annals.2011.173.1.3CrossRefGoogle Scholar
Sixsmith, D. J.. Entire functions for which the escaping set is a spider’s web. Math. Proc. Cambridge Philos. Soc. 151(3) (2011), 551–571.10.1017/S0305004111000582CrossRefGoogle Scholar
Sixsmith, D. J.. Maximally and non-maximally fast escaping points of transcendental entire functions. Math. Proc. Cambridge Philos. Soc. 158(2) (2015), 365–383.10.1017/S0305004115000018CrossRefGoogle Scholar
Sixsmith, D. J.. Dynamical sets whose union with infinity is connected. Ergod. Th. & Dynam. Sys. doi:10.1017/etds.2018.54. Published online 10 August 2018.Google Scholar
Wallin, H.. Metrical characterization of conformal capacity zero. J. Math. Anal. Appl. 58(2) (1977), 298–311.10.1016/0022-247X(77)90208-6CrossRefGoogle Scholar
Warren, L.. On the iteration of quasimeromorphic mappings. Math. Proc. Cambridge Philos. Soc. doi:10.1017/S030500411800052X. Published online 6 July 2018.Google Scholar
Waterman, J.. Slow escape in tracts. Proc. Amer. Math. Soc. 147 (2019), 3087–3101.10.1090/proc/14509CrossRefGoogle Scholar