Commuting finite Blaschke products with no fixed points in the unit disk

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...

The Location of Critical Points of Finite Blaschke Products

A theorem of Bôcher and Grace states that the critical points of a cubic polynomial are the foci of an ellipse tangent to the sides of the triangle joining the zeros. A more general result of Siebert and others states that the critical points of a polynomial of degree N are the algebraic foci of a curve of class N − 1 which is tangent to the lines joining pairs of zeroes. We prove the analogous...

Boundary Interpolation by Finite Blaschke Products

Given 2n distinct points z1, z′ 1, z2, z ′ 2, . . . , zn, z ′ n (in this order) on the unit circle, and n points w1, . . . , wn on the unit circle, we show how to construct a Blaschke product B of degree n such that B(zj) = wj for all j and, in addition, B(z′ j) = B(z ′ k) for all j and k. Modifying this example yields a Blaschke product of degree n− 1 that interpolates the zj ’s to the wj ’s. ...

On random points in the unit disk

Let n be a positive integer and λ > 0 a real number. Let Vn be a set of n points in the unit disk selected uniformly and independently at random. Define G(λ, n) to be the graph with vertex set Vn, in which two vertices are adjacent if and only if their Euclidean distance is at most λ. We call this graph a unit disk random graph. Let λ = c √ lnn/n and let X be the number of isolated points in G(...

Common Fixed Points of Two Commuting Mappings in Complete Metric Spaces