On common fixed points, periodic points, and recurrent points of continuous functions

Abstract

It is known that two commuting continuous functions on an interval need not have a common fixed point. However, it is not known if such two functions have a common periodic point. we had conjectured that two commuting continuous functions on an interval will typically have disjoint sets of periodic points. In this paper, we first prove thatSis a nowhere dense subset of[0,1]if and only if{f∈C([0,1]):Fm(f)∩S¯≠∅}is a nowhere dense subset ofC([0,1]). We also give some results about the common fixed, periodic, and recurrent points of functions. We consider the class of functionsfwith continuousωfstudied by Bruckner and Ceder and show that the set of recurrent points of such functions are closed intervals.