Commuting analytic functions without fixed points

Fixed Points of Analytic 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...

Fixed Points of Commuting Holomorphic Maps without Boundary Regularity

We identify a class of domains of C in which any two commuting holomorphic self-maps have a common fixed point.

Quasi-completeness and functions without fixed-points

Q-reducibility is a natural generalization of many-one reducibility (m-reducibility): if A ≤m B via a computable function f(x), then A ≤Q B via the computable function g(x) such that Wg(x) = {f(x)}. Also this reducibility is connected with enumeration reducibility (e-reducibility) as follows: if A ≤Q B via a computable function g(x), then ω −A ≤e ω −B via the c. e. set W = {〈x, 2y〉 | x ∈ ω, y ∈...

Digital Fixed Points, Approximate Fixed Points, and Universal Functions

A. Rosenfeld [23] introduced the notion of a digitally continuous function between digital images, and showed that although digital images need not have fixed point properties analogous to those of the Euclidean spaces modeled by the images, there often are approximate fixed point properties of such images. In the current paper, we obtain additional results concerning fixed points and approxima...