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 that S is a nowhere dense subset of [ 0 , 1 ] if and only if { f ∈ C ( [ 0 , 1 ] ) : F m ( 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 ω f studied by Bruckner and Ceder and show that the set of recurrent points of such functions are closed intervals.