Abstract
We extend results about the dimension of the radial Julia set of certain exponential functions to quasiregular Zorich maps in higher dimensions. Our results improve on previous estimates of the dimension also in the special case of exponential functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. Barański, B. Karpińska and A. Zdunik, Hyperbolic dimension of Julia sets of meromorphic maps with logarithmic tracts, International Mathematics Research Notices 2009 (2009), 615–624.
W. Bergweiler, Iteration of meromorphic functions, Bulletin of the American Mathematical Society 29 (1993), 151–188.
W. Bergweiler, Karpińska’s paradox in dimension 3, Duke Mathematical Journal 154 (2010), 599–630.
P. Comdühr, On the differentiability of hairs for Zorich maps, Ergodic Theory and Dynamical Systems 39 (2019), 1824–1842.
R. L. Devaney and M. Krych, Dynamics of exp(z), Ergodic Theory and Dynamical Systems 4 (1984), 35–52.
A. È. Erëmenko, On the iteration of entire functions, in Dynamical Systems and Ergodic Theory (Warsaw, 1986), Banach Center Publications, Vol. 23, Polish Scientific Publishers, Warsaw, 1989, pp. 339–345.
K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, John Wiley & Sons, Chichester, 1990.
G. Havard, M. Urbański and M. Zinsmeister, Variations of Hausdorff dimension in the exponential family, Annales Academiæ Scientiarum Fennicæ. Mathematica 35 (2010), 351–378.
T. Iwaniec and G. Martin, Geometric Function Theory and Non-linear Analysis, Oxford Mathematical Monographs, Oxford University Press, New York, 2001.
B. Karpińska, Area and Hausdorff dimension of the set of accessible points of the Julia sets of λez and λ sin z, Fundamenta Mathematicae 159 (1999), 269–287.
B. Karpińska, Hausdorff dimension of the hairs without endpoints for λ exp z, Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 328 (1999), 1039–1044.
O. Martio and U. Srebro, Periodic quasimeromorphic mappings in Rn, Journal d’Analyse Mathématique 28 (1975), 20–40.
C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Transactions of the American Mathematical Society 300 (1987), 329–342.
L. Rempe, Hyperbolic dimension and radial Julia sets of transcendental functions, Proceedings of the American Mathematical Society 137 (2009), 1411–1420.
D. Schleicher, Dynamics of entire functions, in Holomorphic Dynamical Systems, Lecture Notes in Mathematics, Vol. 1998, Springer, Berlin, 2010, pp. 295–339.
M. Urbański and A. Zdunik, The finer geometry and dynamics of the hyperbolic exponential family, Michigan Mathematical Journal 51 (2003), 227–250.
M. Urbański and A. Zdunik, Real analyticity of Hausdorff dimension of finer Julia sets of exponential family, Ergodic Theory and Dynamical Systems 24 (2004), 279–315.
M. Urbański and A. Zdunik, The parabolic map \({f_{1/e}}\left( z \right) = {1 \over e}{e^z}\), Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae 15 (2004), 419–433.
V. A. Zorich, A theorem of M. A. Lavrent’ev on quasiconformal space maps, Matematicheskiĭ Sbornik 74 (1967), 417–433; English translation: Mathematics of the USSR. Sbornik 3 (1967), 389–403.
Acknowledgments
This research was initiated while the second author was visiting the University of Kiel and it was completed while the first author was visiting the Shanghai Center of Mathematical Sciences. Both authors thank the respective institutions for the hospitality.
We also thank the referee for a large number of helpful remarks, in particular for suggesting an improvement of the lower bound in Theorem 1.1.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bergweiler, W., Ding, J. Non-escaping points of Zorich maps. Isr. J. Math. 243, 27–43 (2021). https://doi.org/10.1007/s11856-021-2140-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-021-2140-2