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

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 that S is a nowhere dense subset of [0,1] if and only if { f∩C ([0, 1]):Fm (f)∩S^-□} is a nowhere dense subset of C ([0,1]). We also give some results about the common fixed, periodic, and recurrent points of functions. We consider the class of functions f with continuous ωf studied by Bruckner and Ceder and show that the set of recurrent points of such functions are closed intervals.