Sphere-preserving maps in inversive geometry

We give an extensive discussion of sphere-preserving maps defined on subdomains of Euclidean -space, and their relationship to Möbius maps and to the preservation of cross-ratios. In the case (the complex plane) we also relate these ideas to the solutions of certain functional equations.

     

On Ritt's Factorization of Polynomials

Ritt has shown that any complex polynomial p can be writtenas the composition of polynomials p₁,...,p_m, where each p_j isprime in the sense that it cannot be written as a non-trivialcomposition of polynomials. The factors p_j are not unique butthe number m of them is, as is the set of the degrees of thep_j. The paper extends Ritt's theory and, in particular, a thirdinvariant of the decomposition is introduced.     

The Critical Values of a Polynomial

It has been known for a long time that any real sequence y ₁ , . . . ,y _n-1 is the sequence of critical values of some real polynomial. Here we show that any complex sequence w ₁ , . . . ,w _n-1 is the sequence of critical values of some complex polynomial.     

A remark on analytic continuation

A simple example is given to show that the space of germs obtained by analytic continuation of a given germ need not be a covering space in the topological sense.

     

The Schwarz-Pick Lemma for derivatives

The Schwarz-Pick Lemma states that any analytic function of the unit disc into itself is a contraction with respect to the hyperbolic metric. In this note a related result is proved for the derivative of an analytic function.

     

The Hillam–Thron Theorem in Higher Dimensions

Let D be a disc in the complex plane, and let s _n be a sequence of Möbius maps each of which maps D into itself. In 1965 Hillam and Thron proved (essentially) that if the points s _n (), n=12,..., lie in a compact subset of D then the functions s ₁^...^s _n converge locally uniformly to a constant on D. They then made applications to continued fractions. In this paper, the corresponding results are proved in all dimensions.     

Euclidean and hyperbolic lengths of images of arcs

Let f be a function that is analytic in the unit disc. We givenew estimates, and new proofs of existing estimates, of theEuclidean length of the image under f of a radial segment inthe unit disc. Our methods are based on the hyperbolic geometryof plane domains, and we address some new questions that follownaturally from this approach.