Michael Frame

FRACTAL GEOMETRY OF RESTRICTED SETS OF CIRCLE INVERSIONS

For intersecting circles, we propose a modification of the limit set generated by inversion in circles. This restricted limit set is always a subset of the discs bounded by the generating circles. We give examples of restricted limit sets and show arrangements of generating circles for which the restricted limit set equals the limit set, and also arrangements for which they differ. In addition, we give a visual presentation, based on Iterated Function Systems, of the excluded strings in the restricted limit set. This leads to a graphical representation of the grammatical complexity of the restricted limit set.

Some nonlinear iterated function systems

Abstract We give examples of Iterated Function Systems where the usual affine functions are replaced by complex polynomials. Since these are not global contractions, some care must be exerted with the domain on which the functions are applied. On the other hand, the lack of unique inverse functions gives rise to multiple addresses for the same regions of the attractor, thus providing another degree of information available for determining other attributes—for example, colors or textures—of the image. Traversing a loop through parameter space reveals an “explosion” of features and also what may be a curious cascade of self-intersections.

An infinite circle inversion limit set fractal

Abstract Analogous to the fractals generated by iterated function systems (IFS) are the limit sets of circle inversions. Typically, these are generated by circles that are mutually external, that is, each lies outside all the others. By relaxing this condition, we construct a circle inversion limit set of infinite extent, a fractal that is scale-independent in a stronger sense than the familiar IFS fractals.

WHEN IS A RECURRENT IFS ATTRACTOR A STANDARD IFS ATTRACTOR

Imposing restrictions on the allowed sequences of transformations of a standard IFS can give rise to attractors that are attractors of standard IFS consisting of a (larger) finite set of transformations, or a countably infinite set of transformations, or are not the attractor of any standard IFS. Whichever the case it can be read off from the transition graph of the restrictions.

A generalized mandelbrot set and the role of critical points

Abstract We examine the Julia sets for certain cubic and quartic polynomials and observe the structure of the Julia set as determined by the orbits of the critical points. In examples where the orbits of some critical points diverge and others converge, or where orbits converge to different cycles, we note the local structure of the Julia set is dominated by the behavior of the nearest critical point. This information is encoded in a generalized Mandelbrot set reflecting the behavior of all the critical points.

A PRIMER OF NEGATIVE TEST DIMENSIONS AND DEGREES OF EMPTINESS FOR LATENT SETS

Applied blindly, the formula for the dimension of the intersection can give negative results. Extending Minkowskis definition of the dimension by ∊-neighborhoods to ∊-pseudo-neighborhoods, that is, replacing (A ∩ B)∊ with A∊ ∩ B∊, we introduce the notion of negative dimensions through several examples of random fractal constructions.

FRACTAL VIDEOFEEDBACK AS ANALOG ITERATED FUNCTION SYSTEMS

We demonstrate that videofeedback augmented with one or two mirrors can produce stationary fractal patterns obtained by an iterated function system (IFS) with two transformations (one with a reflection, one without), and with four transformations (two with reflections, one without, one with a 180° rotation). Camera placement yielding only a partial image in one or both mirrors can be achieved using IFS with memory. The IFS rules are obtained from the images of three non-collinear points in each of the principal pieces of the image.

Scaling symmetries in nonlinear dynamics: a view from parameter space

Abstract The orbit of the critical point of a discrete nonlinear dynamical system defines a family of polynomials in the parameter space of that system. We show here that for the important class of quadratic-like maps, these polynomials become indistinguishable under a suitable scaling transformation. The universal representation of these polynomials produced by such a scaling leads directly to accurate approximations for those parameter values where windows of order appear, the sizes of such windows, measures of window distortion, and the characterization of the internal structure of the windows in terms of generalized Feigenbaum numbers.

HIGHER BLOCK IFS 1: MEMORY REDUCTION AND DIMENSION COMPUTATIONS

By applying a result from the theory of subshifts of finite type,1 we generalize the result of Frame and Lanski2 to IFS with multistep memory. Specifically, we show that for an IFS with m-step memory, there is an IFS with 1-step memory (though in general with many more transformations than ) having the same attractor as .

Reverse bifurcations in a quartic family

Abstract We show a family of real quartic maps exhibits reverse bifurcations. Also called cycle-annihilating transitions, reverse bifurcations have been known for the Henon map for several years, but are not present in the logistic map. We use trapping squares to explain these quartic reverse bifurcations.