Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T11:18:22.518Z Has data issue: false hasContentIssue false
  • English
  • Français

The $\omega $-limit sets of quadratic Julia sets

Published online by Cambridge University Press:  27 September 2013

ANDREW D. BARWELL  and
BRIAN E. RAINES
ANDREW D. BARWELL
Affiliation:
Heilbronn Institute of Mathematical Research, University of Bristol, Howard House, Queens Avenue, Bristol BS8 1SN, UK email [email protected] School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK email [email protected]
BRIAN E. RAINES
Affiliation:
Department of Mathematics, Baylor University, Waco, TX 76798-7328, USA email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we characterize $\omega $-limit sets of dendritic Julia sets for quadratic maps. We use Baldwin’s symbolic representation of these spaces as a non-Hausdorff itinerary space and prove that quadratic maps with dendritic Julia sets have shadowing, and also that for all such maps, a closed invariant set is an $\omega $-limit set of a point if, and only if, it is internally chain transitive.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 2 , April 2015 , pp. 337 - 358
Copyright
© Cambridge University Press, 2013 

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

Alsedà, L. and Fagella, N.. Dynamics on Hubbard trees. Fund. Math. 164 (2) (2000), 115–141.Google Scholar
Baldwin, S.. Continuous itinerary functions and dendrite maps. Topology Appl. 154 (16) (2007), 2889–2938.Google Scholar
Baldwin, S.. Inverse limits of tentlike maps on trees. Fund. Math. 207 (3) (2010), 211–254.Google Scholar
Baldwin, S.. Julia sets and periodic kneading sequences. J. Fixed Point Theory Appl. 7 (1) (2010), 201–222.Google Scholar
Barwell, A., Good, C., Knight, R. and Raines, B. E.. A characterization of $\omega $-limit sets in shift spaces. Ergod. Th. & Dynam. Sys. 30 (1) (2010), 21–31.Google Scholar
Barwell, A. D.. A characterization of $\omega $-limit sets of piecewise monotone maps of the interval. Fund. Math. 207 (2) (2010), 161–174.Google Scholar
Barwell, A. D., Good, C., Oprocha, P. and Raines, B. E.. Characterizations of $\omega $-limit sets in topologically hyperbolic systems. Discrete Contin. Dyn. Syst. 33 (5) (2013), 1819–1833.Google Scholar
Barwell, A. D., Davies, G. and Good, C.. On the $\omega $-limit sets of tent maps. Fund. Math. 217 (1) (2012), 35–54.Google Scholar
Douady, A. and Hubbard, J. H.. Étude dynamique des polynômes complexes. Partie I (Publications Mathématiques d’Orsay, 84). Université de Paris-Sud, Département de Mathématiques, Orsay, 1984.Google Scholar
Fine, N. J. and Wilf, H. S.. Uniqueness theorems for periodic functions. Proc. Amer. Math. Soc. 16 (1965), 109–114.Google Scholar
Hirsch, M. W., Smith, H. L. and Zhao, X.-Q.. Chain transitivity, attractivity, and strong repellors for semidynamical systems. J. Dynam. Differential Equations 13 (1) (2001), 107–131.Google Scholar
Schleicher, D.. On fibers and local connectivity of Mandelbrot and Multibrot sets. Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot. Part 1 (Proceedings of Symposia in Pure Mathematics, 72). American Mathematical Society, Providence, RI, 2004, pp. 477–517.Google Scholar
Walters, P.. On the pseudo-orbit tracing property and its relationship to stability. The Structure of Attractors in Dynamical Systems (Proc. Conf., North Dakota State Univ., Fargo, ND, 1977) (Lecture Notes in Mathematics, 668). Springer, Berlin, 1978, pp. 231–244.Google Scholar