Article

Distributional Chaos in Dendritic and Circular Julia Sets

Journal of Mathematical Analysis and Applications
Document Type

Article
Publication Date

8-1-2015
DOI

10.1016/j.jmaa.2015.03.028
Abstract

If x and y belong to a metric space X , we call (x,y) a DC1 scrambled pair for f:X→X if the following conditions hold: 1) for all t>0, , and 2) for some t>0, .
If D⊂X is an uncountable set such that every x,y∈D form a DC1 scrambled pair forf, we say f exhibits distributional chaos of type 1. If there exists t>0 such that condition 2) holds for any distinct points x,y∈D, then the chaos is said to be uniform. A dendrite is a locally connected, uniquely arcwise connected, compact metric space. In this paper we show that a certain family of quadratic Julia sets (one that contains all the quadratic Julia sets which are dendrites and many others which contain circles) has uniform DC1 chaos.
Disciplines

Keywords

- Schweizer–Smítal chaos,
- Distributional chaos,
- DC1,
- Scrambled set,
- Julia set,
- Dendrite

Citation Information

Nathan Averbeck and Brian E. Raines. "Distributional Chaos in Dendritic and Circular Julia Sets" *Journal of Mathematical Analysis and Applications*Vol. 428 Iss. 2 (2015) p. 951 - 958

Available at: http://works.bepress.com/nathan-averbeck/5/