**Real
Projective Iterated Function Systems** - Michael F. Barnsley,
Andrew Vince and David C. Wilson

This paper contains four main results associated with an attractor of a projective iterated function system (IFS). The rst theorem characterizes when a projective IFS has an attractor which avoids a hyperplane. The second theorem establishes that a projective IFS has at most one attractor. In the third theorem the classical duality between points and hyperplanes in projective space leads to connections between attractors that avoid hyperplanes and repellers that avoid points as well as hyperplane attractors that avoid points and repellers that avoid hyperplanes. Finally, an index is dened for attractors which avoid a hyperplane. This index is shown to be a nontrivial projective invariant.

**A
Characterization of Hyperbolic Affine Iterated Function Systems**
- Ross Atkins, Michael F. Barnsley, Andrew Vince, David C. Wilson

The two main theorems of this paper provide a characterization of hyperbolic affine iterated function systems defined on Rm . Atsushi Kameyama (Distances on Topological Self-Similar Sets, Proceedings of Symposia in Pure Mathematics, Volume 72.1, 2004) asked the following fundamental question: given a topological self-similar set, does there exist an associated system of contraction mappings? Our theorems imply an affirmative answer to Kameyama’s question for selfsimilar sets derived from affine transformations on Rm.

arXiv:0908.1416v1 [math.GT]

**Transformations
between Fractals** - Michael F. Barnsley

We observe that there exists a natural homeomorphism between the attractors of any two iterated functiuon systems, with coding maps, that have equivalent address structures. Then we show that a generalized Minkowski metric may be used to establish conditions under which an affine iterated function system is hyperbolic. We use these results to construct families of fractal homeomorphisms on a triangular subset of R2. We also give conditions under which certain bilinear iterated function systems are hyperbolic and use them to generate families of homeomorphisms on the unit square. These families are associated with "tilings" of the unit square by fractal curves, some of whose box-counting dimensions can be given explicitly.

Progress in Probability, Vol 61, Fractal Geometry and Stochastics IV, pp 227-250, Bandt, Christoph; Mörters, Peter; Zähle, Martina (Eds.) Birkhauser 2009

**V
-variable fractals:
Fractals with partial self similarity** - Michael F.
Barnsley, John E. Hutchinson, Örjan Stenflo

We establish properties of a new type of fractal which has partial self similarity at all scales. For any collection of iterated functions systems with an associated probability distribution and any positive integer V there is a corresponding class of V -variable fractal sets or measures. These V -variable fractals can also be obtained from the points on the attractor of a single deterministic iterated function system. Existence, uniqueness and approximation results are established under average contractive assumptions. We also obtain extensions of some basic results concerning iterated function systems.

Advances in Mathematics 218 (2008) 2051–2088

**Transformations
Between Self-Referential Sets** - Michael F. Barnsley

In this essay we introduce fractal transformations. The main examples are fascinating mappings between diverse subsets of R

^{2}; they can be readily illustrated by using the chaos game. Fractal transformations can be quicky grasped because they rely on basic notions in topology, probability, dynamical systems, and geometry. They may be applied to computer graphics to produce digital content with new look-and-feel; they may also be relevant to image compression and biological modelling.

Amer. Math. Monthly 116 (2009) 291-304

**New
Methods in Fractal Imaging** - Michael F. Barnsley, John
Hutchinson

In this paper we draw attention to some recent advances in fractal geometry and point out several ways in which they apply to digital imaging. Simple applications include a method for animating backgrounds in the production of synthetic content, including seascapes, forests, and skies; a novel low-cost technique for creating animated talking heads with unique look-and-feel; and the sharing of engaging graphics, at low bandwidth, between wireless devices such as cellphones. These advances make use of an addressing system which may be associated with the "top" of the attractor of an iterated function system (IFS). Previous computer graphics applications of IFS theory have focused on models based on the attractors and the invariant measures of IFSs. The addressing system enables the establishment of mappings between attractors; it is these transformations, rather than the attractors themselves, that underlie the digital imaging ideas introduced here.

Proceedings of the International Conference on Computer Graphics, Imaging and Visualisation, (July 26-28, 2006) IEEE Society, Washington DC 296-301

**Existence
and Uniqueness of Orbital Measures** - Michael F. Barnsley

We note an elementary proof of the existence and uniqueness of a solution $\mu \in \mathbb{P}(\mathbb{X})$ to the equation $\mu = p \mu_0 + q \hat{F} \mu$. Here $\mathbb{X}$ is a topological space, $\mathbb{P}(\mathbb{X})$ is the set of Borel measures of unit mass on $\mathbb{X}$, $\mu_0 \in \mathbb{P}(\mathbb{X})$ is given, $p > 0$, and $q \ge 0$ with $p + q = 1$. The transformation $\hat{F} : \mathbb{P}(\mathbb{X}) \rightarrow \mathbb{P}(\mathbb{X})$ is defined by $\hat{F} v = \sum_{n = 1}^N p_n v \circ f_n^{-1}$ where $f_n : \mathbb{X} \rightarrow \mathbb{X}$ is continuous, $p_n > 0$ for $n = 1, 2, \ldots, N$, $N$ is a finite strictly positive integer, and $\sum_{n = 1}^N p_n = 1.$ This problem occurs in connection with iterated function systems (IFS).

**Theory and
Applications of Fractal Tops** - Michael Barnsley

We consider an iterated function system (IFS) of one-to-one contractive maps on a compact metric space. We define the top of an IFS; define an associated symbolic dynamical system; present and explain a fast algorithm for computing the top; describe an example in one dimension with a rich history going back to work of A.Rnyi [Representations for Real Numbers and Their Ergodic Properties, Acta Math. Acad. Sci. Hung.,8 (1957), pp. 477- 493]; and we show how tops may be used to help to model and render synthetic pictures in applications in computer graphics.

Fractals in Engineering: New Trends in Theory and Applications , Springer-Verlag (2005) Edited by Jacques Levy-Vehel and Evelyne Lutton. pages 3-20

**A
Fractal Valued Random Iteration Algorithm and Fractal Hierarchy**
- Michael Barnsley, John Hutchinson and Orjan Stenflo

We describe new families of random fractals, referred to as "V-variable", which are intermediate between the notions of deterministic and of standard random fractals. The parameter V describes the degree of "variability" at each magnification level any V-variable fractals has at most V "forms" or "shapes". V-variable random fractals have the surprising property that they can be computed using a forward process. More precisely, a version of the usual Random Iteration Algorithm, operating on sets (or measures) rather than points, can be used to sample each family. To present this theory, we review relevant results on fractals (and fractal measures), both deterministic and random. Then our new results are obtained by constructing an iterated function system (a super IFS) from a collection of standard IFSs together with a corresponding set of probabilities. The attractor of the super IFS is called a superfractal; it is a collection of V-variable random fractals (sets or measures) together with an associated probability distribution on this collection. When the underlying space is for example R2 and the transformations are computationally straightforward (such as affine transformations), the superfractal can be sampled by means of the algorithm, which is highly efficient in terms of memory usage. The algorithm is illustrated by some computed examples. Some variants, special cases, generalizations of the framework, and potential applications are mentioned.

**V-variable
fractals and superfractals** - Michael Barnsley, John
Hutchinson and Orjan Stenflo

Deterministic and random fractals, within the framework of IFS, have been used to model and study a wide range of phenonema across many areas of science and technology. However, for many applications deterministic fractals are locally too similar near distinct points while standard random fractals have too little local correlation. Random fractals are also slow and difficult to compute. These two major problems restricting further applications are solved here by the introduction of V-variable fractals and superfractals.

**Fractal
Transformations** - Michael Barnsley and Louisa Barnsley

A strange game of soccer is used to introduce transformations and fractals. Low Information content geometrical transformations of pictures are considered. Fractal transformations and a new way to render pictures of fractals are introduced. These ideas have applications in digital content creation.

The Colours of Infinity: The Beauty and Power of Fractals’, pp. 66-81. London Clear Books, 2004

**Ergodic
Theory, Fractal Tops and Colour Stealing** - Michael
Barnsley

A new structure that may be associated with IFS and superIFS is described. In computer graphics applications this structure can be rendered using a new algorithm, called the 'colour stealing' algorithm.

Unpublished lecture notes