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Knobbly but nice

Part of: Complex dynamical systems

Published online by Cambridge University Press:  08 May 2020

NEIL DOBBS Open the ORCID record for NEIL DOBBS [Opens in a new window]
NEIL DOBBS*
Affiliation:
School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 14, Ireland email neil.dobbs@ucd.ie
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Abstract

Our main result states that, under an exponential map whose Julia set is the whole complex plane, on each piecewise smooth Jordan curve there is a point whose orbit is dense. This has consequences for the boundaries of nice sets, used in induction methods to study ergodic and geometric properties of the dynamics.

Keywords

low-dimensional dynamicscomplex dynamicstranscendentalexponential

MSC classification

Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 7 , July 2021 , pp. 2016 - 2022
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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References

Dobbs, N.. Nice sets and invariant densities in complex dynamics. Math. Proc. Cambridge Philos. Soc. 150(1) (2011), 157–165.10.1017/S0305004110000265CrossRefGoogle Scholar
Dobbs, N.. Measures with positive Lyapunov exponent and conformal measures in rational dynamics. Trans. Amer. Math. Soc. 364(6) (2012), 2803–2824.CrossRefGoogle Scholar
Dobbs, N.. Line, spiral, dense. Enseign. Math. 62(1–2) (2016), 91–107.CrossRefGoogle Scholar
Dobbs, N.. Dense yet elementary. Math. Intelligencer 39(2) (2017), 55–58.CrossRefGoogle Scholar
Erëmenko, A. È.. 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
Hubbard, J. H.. Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz. Topological Methods in Modern Mathematics (Stony Brook, NY, 1991). Publish or Perish, Houston, TX, 1993, pp. 467–511.Google Scholar
Milnor, J.. Local connectivity of Julia sets: expository lectures. The Mandelbrot Set, Theme and Variations (London Mathematical Society Lecture Note Series, 274) . Cambridge University Press, Cambridge, 2000, pp. 67–116.CrossRefGoogle Scholar
Przytycki, F. and Rivera-Letelier, J.. Statistical properties of topological Collet–Eckmann maps. Ann. Sci. Éc. Norm. Supér. (4) 40(1) (2007), 135–178.CrossRefGoogle Scholar
Przytycki, F. and Rivera-Letelier, J.. Nice inducing schemes and the thermodynamics of rational maps. Commun. Math. Phys. 301(3) (2011), 661–707.Google Scholar
Rivera-Letelier, J.. A connecting lemma for rational maps satisfying a no-growth condition. Ergod. Th. & Dynam. Sys. 27(2) (2007), 595–636.CrossRefGoogle Scholar
Rivera-Letelier, J. and Shen, W.. Statistical properties of one-dimensional maps under weak hyperbolicity assumptions. Ann. Sci. Éc. Norm. Supér. (4) 47(6) (2014), 1027–1083.CrossRefGoogle Scholar
Roesch, P.. On local connectivity for the Julia set of rational maps: Newton’s famous example. Ann. of Math. (2) 168(1) (2008), 127–174.10.4007/annals.2008.168.127CrossRefGoogle Scholar
Viana da Silva, M.. The differentiability of the hairs of exp(Z). Proc. Amer. Math. Soc. 103(4) (1988), 1179–1184.Google Scholar