Science and Mathematics Faculty Publications

Title

Distributional Chaos in Dendritic and Circular Julia Sets

Document Type

Article

Publication Date

8-2015

Journal Title

Journal of Mathematical Analysis and Applications

Volume

428

Issue

2

First Page

951

Last Page

958

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

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