John H. Elton

Recurrent iterated function systems

Recurrent iterated function systems generalize iterated function systems as introduced by Barnsley and Demko [BD] in that a Markov chain (typically with some zeros in the transition probability matrix) is used to drive a system of mapswj:K →K,j=1, 2,⋯,N, whereK is a complete metric space. It is proved that under “average contractivity,” a convergence and ergodic theorem obtains, which extends the results of Barnsley and Elton [BE]. It is also proved that a Collage Theorem is true, which generalizes the main result of Barnsleyet al. [BEHL] and which broadens the class of images which can be encoded using iterated map techniques. The theory of fractal interpolation functions [B] is extended, and the fractal dimensions for certain attractors is derived, extending the technique of Hardin and Massopust [HM]. Applications to Julia set theory and to the study of the boundary of IFS attractors are presented.

An ergodic theorem for iterated maps

Consider a Markov process on a locally compact metric space arising from iteratively applying maps chosen randomly from a finite set of Lipschitz maps which, on the average, contract between any two points (no map need be a global contraction). The distribution of the maps is allowed to depend on current position, with mild restrictions. Such processes have unique stationary initial distribution [ BE ], [ BDEG ]. We show that, starting at any point, time averages along trajectories of the process converge almost surely to a constant independent of the starting point. This has applications to computer graphics.

Approximation of Measures by Markov Processes and Homogeneous Affine Iterated Function Systems

AbstractIt is shown that under certain conditions, attractive invariant measures for iterated function systems (indeed for Markov processes on locally compact spaces) depend continuously on parameters of the system.We discuss a special class of iterated function systems, the homogeneous affine ones, for which an inverse problem is easily solved in principle by Fourier transform methods. We show that a probability measureμ onRn can be approximated by invariant measures for finite iterated function systems of this class if

A Generalization of Lyapounov's Convexity Theorem to Measures with Atoms

\hat \mu (t)/\hat \mu (a^T t)

Comparison of stop rule and maximum expectations for finite sequences of exchangeable random variables

is positive definite for some nonzero contractive linear mapa:Rn →Rn. Moments and cumulants are also discussed.

Fractal Image Encoding

On considere une inegalite qui affine et quantifie un theoreme de decoupe du a Urbanik (1955) et Dubins et Spanier (1961)

Hidden variable fractal interpolation functions

The distance from the convex hull of the range of an n-dimensional vector-valued measure to the range of that measure is no more than an/2, where a is the largest (one-dimensional) mass of the atoms of the measure. The case a = 0 yields Lyapounovs Convexity Theorem; applications are given to the bisection problem and to the bang-bang principle of optimal control theory.

A new class of markov processes for image encoding

The conclusion of the classical ham sandwich theorem of Banach and Steinhaus may be strengthened: there always exists a common bisecting hyperplane that touches each of the sets, that is, intersects the closure of each set. Hence, if the knife is smeared with mayonnaise, a cut can always be made so that it will not only simultaneously bisect each of the ingredients, but it will also spread mayonnaise on each. A discrete analog of this theorem says that n finite nonempty sets in n-dimensional Euclidean space can always be simultaneously bisected by a single hyperplane that contains at least one point in each set. More generally, for n compactlysupported positive finite Borel measures in Euclidean n-space, there is always a hyperplane that bisects each of the measures and intersects the support of each measure.