Edward Crane

The simple harmonic urn.

We study a generalized Polya urn model with two types of ball. If the drawn ball is red, it is replaced together with a black ball, but if the drawn ball is black it is replaced and a red ball is thrown out of the urn. When only black balls remain, the roles of the colors are swapped and the process restarts. We prove that the resulting Markov chain is transient but that if we throw out a ball every time the colors swap, the process is recurrent. We show that the embedded process obtained by observing the number of balls in the urn at the swapping times has a scaling limit that is essentially the square of a Bessel diffusion. We consider an oriented percolation model naturally associated with the urn process, and obtain detailed information about its structure, showing that the open subgraph is an infinite tree with a single end. We also study a natural continuous-time embedding of the urn process that demonstrates the relation to the simple harmonic oscillator; in this setting, our transience result addresses an open problem in the recurrence theory of two-dimensional linear birth and death processes due to Kesten and Hutton. We obtain results on the area swept out by the process. We make use of connections between the urn process and birth–death processes, a uniform renewal process, the Eulerian numbers, and Lamperti’s problem on processes with asymptotically small drifts; we prove some new results on some of these classical objects that may be of independent interest. For instance, we give sharp new asymptotics for the first two moments of the counting function of the uniform renewal process. Finally, we discuss some related models of independent interest, including a “Poisson earthquakes” Markov chain on the homeomorphisms of the plane.

Extremal Polynomials in Smale’s Mean Value Conjecture

Let p be a non-linear complex polynomial in one variable. Smale’s mean value conjecture is a precise estimate of the derivative p′(z) in terms of the gradients of chords between z and a stationary point on the graph of p. The problem is to determine the correct constant in the estimate, but despite the apparent simplicity of the problem only a small amount of progress has been made since Stephen Smale first posed it in 1981. In this paper we establish the existence of extremal polynomials for Smale’s mean value conjecture, and establish a geometric property of the extremals.

Conical limit sets and continued fractions

Inspired by questions of convergence in continued fraction theory, Erdős, Piranian and Thron studied the possible sets of divergence for arbitrary sequences of Mobius maps acting on the Riemann sphere, S 2 . By identifying S 2 with the boundary of three-dimensional hyperbolic space, H 3 , we show that these sets of divergence are precisely the sets that arise as conical limit sets of subsets of H 3 . Using hyperbolic geometry, we give simple geometric proofs of the theorems of Erdős, Piranian and Thron that generalise to arbitrary dimensions. New results are also obtained about the class of conical limit sets; for example, it is closed under locally quasisymmetric homeomorphisms. Applications are given to continued fractions.

Mean value conjectures for rational maps

Let p be a polynomial in one complex variable. Smales mean value conjecture estimates |p′(z)| in terms of the gradient of a chord from (z, p(z)) to some stationary point on the graph of p. The conjecture does not immediately generalize to rational maps since its formulation is invariant under the group of affine maps, not the full Möbius group. Here we give two possible generalizations to rational maps, both of which are Möbius invariant. In both cases we prove a version with a weaker constant, in parallel to the situation for Smales mean value conjecture. Finally, we discuss some candidate extremal rational maps, namely rational maps all of whose critical points are fixed points.

Circle packing with generalized branching

There is a fairly comprehensive theory of discrete analytic functions based on circle packing. In this theory, discrete analytic functions are represented as maps between circle packings that share combinatorial tangency patterns. Branching behavior, however, has until now been restricted by the need to place branch points at circle centers. In this paper, the authors introduce mechanisms for generalized branching which remove this restriction. Their use is illustrated by overcoming combinatorial obstructions to branching in the construction of a discrete Ahlfors function.

Intrinsic circle domains

Using quasiconformal mappings, we prove that any Riemann surface of finite connectivity and finite genus is conformally equivalent to an intrinsic circle domain U in a compact Riemann surface S. This means that each connected component B of S \ U is either a point or a closed geometric disc with respect to the complete constant curvature conformal metric of the Riemann surface (U union B). Moreover the pair (U,S) is unique up to conformal isomorphisms. We give a generalization to countably infinite connectivity. Finally we show how one can compute numerical approximations to intrinsic circle domains using circle packings and conformal welding.

Relative Riemann mapping criteria and hyperbolic convexity

Let R be a simply-connected Riemann surface with a simplyconnected subdomain U . We give a criterion in terms of conformal reflections to determine whether R can be embedded in the complex plane so that U is mapped onto a disc. If it can, then U is convex with respect to the hyperbolic metric of R, by a theorem of Jørgensen. We discuss the close relationship of our criterion to two generalizations of Jørgensen’s theorem by Minda and Solynin. We generalize our criterion to the quasiconformal setting and also give a criterion for the multiply-connected case, where an embedding is sought that maps a given subdomain onto a circle domain. Let R be a simply-connected hyperbolic Riemann surface. It has a unique complete conformal metric of constant curvature−1, which we call the hyperbolic metric of R. (Some authors work with curvature −4, but this choice does not affect any of our arguments as we shall only be comparing hyperbolic metrics.) We say that a subset D ⊂ Ω is hyperbolically convex with respect to R if for any two points z = w in D, the unique geodesic segment joining z to w in the hyperbolic metric of R is contained in D. More generally, if R is a possibly multiply-connected hyperbolic Riemann surface, we will say that a subset D is weakly hyperbolically convex with respect to R if each connected component of π−1(D) is hyperbolically convex with respect to the unit disc D, where π : D → R is an unbranched analytic universal covering map. If π−1(D) is connected and hyperbolically convex, we will say that D is strongly hyperbolically convex. Flinn [2] defined D to be h-convex if for any two points z, w in D, any shortest geodesic segment in R joining z to w is contained in D. This implies that D is weakly hyperbolically convex with respect to Ω, but not that it is strongly hyperbolically convex. Let D be an open disc or half-plane contained in a hyperbolic plane domain Ω. Vilhelm Jørgensen [4] showed that D is h-convex in Ω. A plane domain D has the universal h-convexity property (UHP) if D is hconvex with respect to each simply-connected hyperbolic plane domain Ω strictly containing D, and there is at least one such Ω. This definition is from Flinn [2], where it is shown that discs and half-planes are the only plane domains with the UHP. Alan Beardon observed to the author that the property of being a Euclidean Received by the editors November 12, 2010 and, in revised form, February 15, 2011. 2010 Mathematics Subject Classification. Primary 30C35, 30C62; Secondary 52A55.

A note on the Hayman-Wu theorem

The Hayman-Wu Theorem states that the preimage of a line or circle L under a conformal mapping from the unit disc D to a simply-connected domain Ω has total Euclidean length bounded by an absolute constant. The best possible constant is known to lie in the interval [π2, 4π), thanks to work of Øyma and Rohde. Earlier, Brown Flinn showed that the total length is at most π2 in the special case in which L ⊂ Ω. Let r be the anti-Möbius map that fixes L pointwise. In this note we extend the sharp bound π2 to the case where each connected component of Ω ∩ r(Ω) is bounded by one arc of ∂Ω and one arc of r (∂Ω). We also strengthen the bounds slightly by replacing Euclidean length with the strictly larger spherical length on D.

On Keogh’s Length Estimate for Bounded Starlike Functions

For a bounded starlike function ƒ on the unit disc, we consider L(r), the length of the image of the circle ¦z¦ = r. Keogh showed that L(r) = O(log 1/(1 − r) as r → 1 and Hayman showed that this is the correct asymptotic. We give an alternative geometric construction which strengthens Hayman’s result, showing that the constant in Keogh’s original inequality is sharp. The analysis uses standard estimates on the hyperbolic metric of plane domains. The self-similarity of the construction allows for the examples to be expressed analytically. For context, we give a brief survey of related estimates on integral means and coefficients of univalent functions.