## Science and Mathematics Faculty Publications

# 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:

If D⊂X is an uncountable set such that every x,y∈D form a DC1 scrambled pair for*f*, 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

## Recommended Citation

Averbeck, Nathan and Raines, Brian E., "Distributional Chaos in Dendritic and Circular Julia Sets" (2015). *Science and Mathematics Faculty Publications*. 353.

https://digitalcommons.cedarville.edu/science_and_mathematics_publications/353