Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T01:00:32.337Z Has data issue: false hasContentIssue false
  • English
  • Français

Dimensions of slowly escaping sets and annular itineraries for exponential functions

Published online by Cambridge University Press:  13 April 2015

D. J. SIXSMITH Open the ORCID record for D. J. SIXSMITH [Opens in a new window]
D. J. SIXSMITH*
Affiliation:
Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the iteration of functions in the exponential family. We construct a number of sets, consisting of points which escape to infinity ‘slowly’, and which have Hausdorff dimension equal to $1$. We prove these results by using the idea of an annular itinerary. In the case of a general transcendental entire function we show that one of these sets, the uniformly slowly escaping set, has strong dynamical properties and we give a necessary and sufficient condition for this set to be non-empty.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 7 , October 2016 , pp. 2273 - 2292
Copyright
© Cambridge University Press, 2015 

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.. Iteration of meromorphic functions. Bull. Amer. Math. Soc. (N.S.) 29(2) (1993), 151–188.CrossRefGoogle Scholar
Bergweiler, W.. Invariant domains and singularities. Math. Proc. Cambridge Philos. Soc. 117(3) (1995), 525–532.CrossRefGoogle Scholar
Bergweiler, W. and Hinkkanen, A.. On semiconjugation of entire functions. Math. Proc. Cambridge Philos. Soc. 126(3) (1999), 565–574.Google Scholar
Bergweiler, W. and Karpińska, B.. On the Hausdorff dimension of the Julia set of a regularly growing entire function. Math. Proc. Cambridge Philos. Soc. 148(3) (2010), 531–551.Google Scholar
Bergweiler, W. and Peter, J.. Escape rate and Hausdorff measure for entire functions. Math. Z. 274(1–2) (2013), 551–572.CrossRefGoogle Scholar
Bishop, C. J.. A transcendental Julia set of dimension $1$ . Preprint, 2011, http://www.math.sunysb.edu/∼bishop/papers/.Google Scholar
Bodelón, C., Devaney, R. L., Hayes, M., Roberts, G., Goldberg, L. R. and Hubbard, J. H.. Hairs for the complex exponential family. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9(8) (1999), 1517–1534.Google Scholar
Devaney, R. L. and Krych, M.. Dynamics of exp(z). Ergod. Th. & Dynam. Sys. 4(1) (1984), 35–52.Google Scholar
Devaney, R. L. and Tangerman, F.. Dynamics of entire functions near the essential singularity. Ergod. Th. & Dynam. Sys. 6(4) (1986), 489–503.Google Scholar
Eremenko, A. E.. On the iteration of entire functions. Dynamical Systems and Ergodic Theory (Warsaw 1986) (Banach Center Publications, 23) . Polish Scientific Publishers, Warsaw, 1989, pp. 339–345.Google Scholar
Eremenko, A. E. and Lyubich, M. Y.. Examples of entire functions with pathological dynamics. J. Lond. Math. Soc. (2) 36(3) (1987), 458–468.Google Scholar
Eremenko, A. E. and Lyubich, M. Y.. Dynamical properties of some classes of entire functions. Ann. Inst. Fourier (Grenoble) 42(4) (1992), 989–1020.Google Scholar
Falconer, K.. Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. Wiley, Chichester, 2006.Google Scholar
Karpińska, B.. Area and Hausdorff dimension of the set of accessible points of the Julia sets of 𝜆e z and 𝜆sinz . Fund. Math. 159(3) (1999), 269–287.Google Scholar
Karpińska, B.. Hausdorff dimension of the hairs without endpoints for 𝜆expz . C. R. Acad. Sci., Paris 328(11) (1999), 1039–1044.Google Scholar
Karpińska, B. and Urbański, M.. How points escape to infinity under exponential maps. J. Lond. Math. Soc. (2) 73(1) (2006), 141–156.Google Scholar
McMullen, C.. Area and Hausdorff dimension of Julia sets of entire functions. Trans. Amer. Math. Soc. 300(1) (1987), 329–342.Google Scholar
Pawelec, L. and Zdunik, A.. Indecomposable continua in exponential dynamics-Hausdorff dimension. Preprint 2014, arXiv:1405.7784v1.Google Scholar
Rempe, L. and Stallard, G. M.. Hausdorff dimensions of escaping sets of transcendental entire functions. Proc. Amer. Math. Soc. 138(5) (2010), 1657–1665.Google Scholar
Rippon, P. and Stallard, G. M.. Annular itineraries for entire functions. Trans. Amer. Math. Soc. 367(1) (2015), 377–399.Google Scholar
Rippon, P. J. and Stallard, G. M.. Slow escaping points of meromorphic functions. Trans. Amer. Math. Soc. 363(8) (2011), 4171–4201.Google Scholar
Rippon, P. J. and Stallard, G. M.. Fast escaping points of entire functions. Proc. Lond. Math. Soc. (3) 105(4) (2012), 787–820.Google Scholar
Sixsmith, D. J.. Julia and escaping set spiders’ webs of positive area. IMRN Int. Math. Res. Notices (2014), to appear, doi:10.1093/imrn/rnu245.Google Scholar
Stallard, G. M.. Dimensions of Julia sets of transcendental meromorphic functions. Transcendental Dynamics and Complex Analysis (London Mathametical Society Lecture Note Series, 348) . Cambridge University Press, Cambridge, 2008, pp. 425–446.Google Scholar
Urbański, M. and Zdunik, A.. The finer geometry and dynamics of the hyperbolic exponential family. Michigan Math. J. 51(2) (2003), 227–250.Google Scholar