For a post-critically finite branched covering of the sphere that is a subdivision map of a finite subdivision rule, we define non-expanding spines which determine the existence of a Levy cycle in a non-exhaustive semi-decidable algorithm. Especially when a finite subdivision rule has polynomial growth of edge subdivisions, the algorithm terminates very quickly, and the existence of a Levy cycle is equivalent to the existence of a Thurston obstruction. To show the equivalence between Levy and Thurston obstructions, we generalize the arcs intersecting obstruction theorem by Pilgrim and Tan [Combining rational maps and controlling obstructions. Ergod. Th. & Dynam. Sys. 18(1) (1998), 221–245] to a graph intersecting obstruction theorem. As a corollary, we prove that for a pair of post-critically finite polynomials, if at least one polynomial has core entropy zero, then their mating has a Levy cycle if and only if the mating has a Thurston obstruction.

We prove analogues of several well-known results concerning rational maps between quadrics for the class of so-called quasilinear p-hypersurfaces. These hypersurfaces are nowhere smooth over the base field, so many of the geometric methods which have been successfully applied to the study of projective homogeneous varieties over fields cannot be used. We are therefore forced to take an alternative approach, which is partly facilitated by the appearance of several non-traditional features in the study of these objects from an algebraic perspective. Our main results were previously known for the class of quasilinear quadrics. We provide new proofs here, because the original proofs do not immediately generalise for quasilinear hypersurfaces of higher degree.

Let X be an algebraic variety and let f:X−−→X be a rational self-map with a fixed point q, where everything is defined over a number field K. We make some general remarks concerning the possibility of using the behaviour of f near q to produce many rational points on X. As an application, we give a simplified proof of the potential density of rational points on the variety of lines of a cubic fourfold, originally proved by Claire Voisin and the first author in 2007.

We study the dynamics of a family of third-order iterative methods that are used to find roots of nonlinear equations applied to complex polynomials of degrees three and four. This family includes, as particular cases, the Chebyshev, the Halley and the super-Halleyroot-finding algorithms, as well as the so-called c-methods. The conjugacy classes of theseiterative methods are found explicitly.

We study dynamics of rational maps of degree at least 2 with coefficients in the field ${\open C}_{p}$, where p>1 is a fixed prime number. The main ingredient is to consider the action of rational maps in p-adic hyperbolic space, denoted ${\open H}_{p}$. Hyperbolic space ${\open H}_{p}$ is provided with a natural distance, for which it is connected and one dimensional (an ${\open R}$-tree). These advantages with respect to ${\open C}_{p}$ give new insight into dynamics. In this paper we prove the following results about periodic points; we give applications to the Fatou/Julia theory over ${\open C}_{p}$ and to ultrametric analysis in forthcoming papers. We prove that the existence of at least two nonrepelling periodic points implies the existence of infinitely many of them. This is in contrast with the complex setting where a rational map can have at most finitely many nonrepelling periodic points. On the other hand we prove that every rational map has a repelling fixed point, either in the projective line or in hyperbolic space. So the topological expansion of a rational map is detected by some fixed point.

This paper describes a procedure for the construction of monopoles on three-dimensional Euclidean space, starting from their rational maps. A companion paper, ‘Euclidean monopoles and rational maps’, to appear in the same journal, describes the assignment to a monopole of a rational map, from $\Bbb{CP}^1$ to a suitable flag manifold. In describing the reverse direction, this paper completes the proof of the main theorem therein.

A construction of monopoles from solutions to Nahm's equations (a system of ordinary differential equations) has been well-known for certain gauge groups for some time. These solutions are hard to construct however, and the equations themselves become increasingly unwieldy when the gauge group is not $\mbox{SU}(2).$

Here, in contrast, a rational map is the only initial data. But whereas one can be reasonably explicit in moving from Nahm data to a monopole, here the monopole is only obtained from the rational map after solving a partial differential equation.

A non-linear flow equation, essentially just the path of steepest descent down the Yang-Mills-Higgs functional, is set up. It is shown that, starting from an ‘approximate monopole’ - constructed explicitly from the rational map - a solution to the flow must exist, and converge to an exact monopole having the desired rational map.

1991 Mathematics Subject Classification: 53C07, 53C80, 58D27, 58E15, 58G11.