Skip to main content
Article
Distributional Chaos in Dendritic and Circular Julia Sets
Journal of Mathematical Analysis and Applications
  • Nathan Averbeck, Cedarville University
  • Brian E. Raines
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.
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/