Distributional chaos in dendritic and circular Julia sets

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, limsupn→∞1n|{0≤i<n:d(fi(x),fi(y))<t}|=1, and2)for some t>0, liminfn→∞1n|{0≤i<n:d(fi(x),fi(y))<t}|=0. 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.