A Note on Periodic Points and Commuting Functions

Abstract

Alikhani-Koopaei has recently conjectured that two commuting continuous functions typically share no periodic points. After discussing the history behind Alikhani-Koopaei's conjecture, we use the Baire Category Theorem to investigate the likelihood that two commuting continuous functions have disjoint sets of periodic points, as well as the structure of the set $\mathcal{F}=\{g\in C(I,I):gf=fg\}$. We then turn our attention to the iterative properties possessed by commuting pairs of continuous functions.