Continuity of the mapsf↦∪x∈Iω(x,f)andf↦{ω(x,f):x∈I}

Abstract

We study the behavior of two maps in an effort to better understand the stability ofω-limit setsω(x,f)as we perturb eitherxorf, or both. The first map is the set-valued functionΛtakingfinC(I,I)to its collection ofω-limit pointsΛ(f)=∪x∈Iω(x,f), and the second is the mapΩtakingfinC(I,I)to its collection ofω-limit setsΩ(f)={ω(x,f):x∈I}. We characterize those functionsfinC(I,I)at which each of our mapsΛandΩis continuous, and then go on to show that bothΛandΩare continuous on a residual subset ofC(I,I). We then investigate the relationship between the continuity ofΛandΩat some functionfinC(I,I)with the chaotic nature of that function.