William J. Gilbert

Extraneous fixed points, basin boundaries and chaotic dynamics for Schro¨der and Ko¨nig rational iteration functions

SummaryThe Schröder and König iteration schemes to find the zeros of a (polynomial) functiong(z) represent generalizations of Newtons method. In both schemes, iteration functionsfm(z) are constructed so that sequenceszn+1=fm(zn) converge locally to a rootz* ofg(z) asO(|zn−z*|m). It is well known that attractive cycles, other than the zerosz*, may exist for Newtons method (m=2). Asm increases, the iteration functions add extraneous fixed points and cycles. Whether attractive or repulsive, they affect the Julia set basin boundaries. The König functionsKm(z) appear to minimize such perturbations. In the case of two roots, e.g.g(z)=z2−1, Cayleys classical result for the basins of attraction of Newtons method is extended for allKm(z). The existence of chaotic {zn} sequences is also demonstrated for these iteration methods.

Radix representations of quadratic fields

Abstract Let ϱ be an algebraic integer in a quadratic number field whose minimum polynomial is x 2 + p 1 + p 0 . Then all the elements of the ring |Z [ϱ] can be written uniquely in the base ϱ as Σ k m =0 a kπ k , where 0 ⩽ a k p 0 |, if and only if p 0 ⩾ 2 and −1 ⩽ p 1 ⩽ p 0 .

Fractal geometry derived from complex bases

Each complex number can be expressed as a single number in positional notation using certain complex bases, just as the positive real numbers can be expressed as decimal expansions. These representations yield some intriguing geometric patterns in the complex plane, whose boundaries are fractal curves. One of these curves is known from the investigation of dragon curves; the others are new examples of fractals.

Automatic maps in exotic numeration systems

We generalize the classical notion of ab-automatic sequence for a sequence indexed by the natural numbers. We replace the integers by a semiring and use a numeration system consisting of the powers of a baseb and an appropriate set of digits. For example, we define (−3)-automatic sequences (indexed by the ordinary integers or by the rational integers) and (−1 +i)-automatic sequences (indexed by the Gaussian integers). We show how these new notions are related to the old ones, and we study both the number-theoretic and automata-theoretic properties that permit the replacement of one numeration system by another.

The complex dynamics of Newton's method for a double root

We show that the relaxed Newtons method for finding the roots of a cubic with one double root is conjugate, by a linear fractional transformation on the P,.iemmm sphere, to the iterations of the quadratic p(z) ~- z 2 - 0.75.

Geometry of Radix Representations

The aim of this paper is to illuminate the connection between the geometry and the arithmetic of the radix representations of the complex numbers and other algebraic number fields. We indicate how these representations yield a variety of naturally defined fractal curves and surfaces of higher dimensions.

Newton's method for multiple roots

Abstract We investigate the basins of attraction in the complex plane of Newtons method for finding multiple roots and illustrate what happens as two simple roots coalesce to form a double root.

The fractal dimension of sets derived from complex bases

Determination de la dimension fractale de la frontiere des ensembles derives des bases complexes

The division algorithm in complex bases

Complex numbers can be represented in positional notation using certain Gaussian integers as bases and digit sets. We describe a long division algorithm to divide one Gaussian integer by another, so that the quotient is a periodic expansion in such a complex base. To divide by the Gaussian integer w in the complex base b, using a digit set D, the remainder must be in the set wT(b, D) \ Z[i], where T(b, D) is the set of complex numbers with zero integer part in the base. The set T(b, D) tiles the plane, and can be described geometrically as the attractor of an iterated function system of linear maps. It usually has a fractal boundary. The remainder set can be determined algebraically from the cycles in a certain directed graph. 1. Complex bases. A Gaussian integer b, together with a digit set D, of Gaussian integers containing zero, is called a valid base for the complex numbers if every Gaussian integer, z, can be represented uniquely in the form

Bricklaying and the Hermite normal form

An integer lattice in n-dimensional space is the set of all integer linear combinations of n linearly independent vectors with integer entries. The translation group of the lattice consists of all the translations of n-dimensional space that take lattice points to lattice points. This group is isomorphic to the set of lattice points under vector addition. A fundamental brick will be a rectangular brick with one vertex at the origin and sides along each positive coordinate axis. A brickwork will consist of a tiling of n-space by n-dimensional bricks that are translations of a fundamental brick. The set of these translations form the translation group of the brickwork.