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 that S is a nowhere dense subset of [0, 1] if and only if 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.
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