Common periodic points of commuting functions.

Commuting Functions with No Common Fixed Points)

Introduction. Let/and g be continuous functions mapping the unit interval / into itself which commute under functional composition, that is, f(g(x)) = g(f(x)) for all x in /. In 1954 Eldon Dyer asked whether/and g must always have a common fixed point, meaning a point z in / for which f(z) = z=g(z). A. L. Shields posed the same question independently in 1955, as did Lester Dubins in 1956. The p...

Periodic Points of Functions

By a counter example we show that two continuous functions defined on a compact metric space satisfying a certain semi metric need not have a common periodic point.

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...

Common Periodic Points for a Class of Continuous Commuting Mappings on an Interval

The existence of common periodic points for a family of continuous commuting self-mappings on an interval is proved and two illustrative examples are given in support of our theorem and definition. 1. Introduction and preliminaries. All mappings considered here are assumed to be continuous from the interval I = [u, v] to itself. Let F(f) and P (f) be the set of fixed and periodic points of f , ...

Common Fixed Points of Two Commuting Mappings in Complete Metric Spaces