arXiv subjects

Michael Barnsley

Publications and source records attributed to Michael Barnsley.

18 recordsLinked to original sources

arXiv2023

Remembrances of Derek William Robinson (25 June 1935 - 31 August 2021)

This arXiv submission is posted primarily to make available some additional material to what we prepared for the Derek Memorial article published in the Notices of the AMS. We start (with the permission of the AMS) with the text of our submission close to the final published version (which you can get online. After that we include some autobiographical notes that Derek dictated to Louisa Barnsley (whom we thank) during his final illness. Finally some unpublished notes Derek wrote on the history of his seminal monographs with Bratelli. We thank Marion Robinson for permission and encouragement to post these items.

math.HO↗

arXiv2021

Distortion Reversal in Aperiodic Tilings

It is proved that homeomorphic images of certain two-dimensional aperiodic tilings, such as Ammann-A2 tilings, are recognizable, in both mathematical and practical senses. One implication of the results is that it is possible to search for distorted aperiodic structures in nature, where they may be hiding in plain sight.

math.DS↗

arXiv2021

Central Open Sets Tilings

We introduce a method for constructing collections of subsets of $\mathbb{R}^{n}$, using an iterated function system, a set $T,$ and a cost function. We refer to these collections as tilings. The special case where $T$ is the central open set of an iterated function system that obeys the open set condition is emphasized. The notion of the central open set associated with an iterated function system of similitudes, introduced in 2005 by Bandt, Hung, and Rao, is reviewed. A practical method for calculating pictures of central open sets is described. Some general properties and examples of the tilings are presented.

math.DS↗

arXiv2019

Tilings from Graph Directed Iterated Function Systems

A new method for constructing self-referential tilings of Euclidean space from a graph directed iterated function system, based on a combinatorial structure we call a pre-tree, is introduced. In the special case that we refer to as balanced, the resulting tilings have a finite set of prototiles, are quasiperiodic but not periodic, and are self-similar. A necessary and sufficient condition for two balanced tilings to be congruent is provided.

math.MG↗

arXiv2016

Self-Similar Polygonal Tiling

The purpose of this paper is to give the flavor of the subject of self-similar tilings in a relatively elementary setting, and to provide a novel method for the construction of such polygonal tilings.

math.MG↗

arXiv2016

Self-referential Functions

We introduce the concept of fractels for functions and discuss their analytic and algebraic properties. We also consider the representation of polynomials and analytic functions using fractels, and the consequences of these representations in numerical analysis.

math.CA↗

arXiv2014

Conjugacies provided by fractal transformations I : Conjugate measures, Hilbert spaces, orthogonal expansions, and flows, on self-referential spaces

Theorems and explicit examples are used to show how transformations between self-similar sets (general sense) may be continuous almost everywhere with respect to stationary measures on the sets and may be used to carry well known flows and spectral analysis over from familiar settings to new ones. The focus of this work is on a number of surprising applications including (i) what we call fractal Fourier analysis, in which the graphs of the basis functions are Cantor sets, being discontinuous at a countable dense set of points, yet have very good approximation properties; (ii) Lebesgue measure-preserving flows, on polygonal laminas, whose wave-fronts are fractals. The key idea is to exploit fractal transformations to provide unitary transformations between Hilbert spaces defined on attractors of iterated function systems. Some of the examples relate to work of Oxtoby and Ulam concerning ergodic flows on regions bounded by polygons.

math.DS↗

arXiv2014

A Combinatorial Characterization of the Critical Itineraries of a Uniform Dynamical System

For a function from the unit interval to itself with constant slope and one discontinuity, the itineraries of the point of discontinuity are called the critical itineraries. These critical itineraries play a significant role in the study of $β$-expansions (with positive or negative $β$) and fractal transformations. A combinatorial characterization of the critical itineraries of such functions is provided.

math.DS↗

arXiv2013

Fractal Tiling

A simple, yet unifying method is provided for the construction of tilings by tiles obtained from the attractor of an iterated function system (IFS). Many examples appearing in the literature in ad hoc ways, as well as new examples, can be constructed by this method. These tilings can be used to extend a fractal transformation defined on the attractor of a contractive IFS to a fractal transformation on the entire space upon which the IFS acts.

math.MG↗

arXiv2013

Fractal Transformations in 2 and 3 Dimensions

We present some work relating to fractal transformations on masked iterated function systems and demonstrate how well known algorithms for generating fractal transformations can be modifed for these systems. We also demonstrate that these algorithms work equally well when applied to three dimensional data sets and suggest some possible applications to special effects and modelling.

math.DS↗

arXiv2012

The Conley Attractor of an Iterated Function System

We investigate the topological and metric properties of attractors of an iterated function system (IFS) whose functions may not be contractive. We focus, in particular, on invertible IFSs of finitely many maps on a compact metric space. We rely on ideas Kieninger and McGehee and Wiandt, restricted to what is, in many ways, a simpler setting, but focused on a special type of attractor, namely point-fibred minimal (locally) invariant sets. This allows us to give short proofs of some of the key ideas.

math.DS↗

arXiv2011

Fractal Homeomorphism for Bi-affine Iterated Function Systems

The paper concerns fractal homeomorphism between the attractors of two bi-affine iterated function systems. After a general discussion of bi-affine functions, conditions are provided under which a bi-affine iterated function system is contractive, thus guaranteeing an attractor. After a general discussion of fractal homeomorphism, fractal homeomorphisms are constructed for a specific type of bi-affine iterated function system.

math.DS↗

arXiv2010

The Chaos Game on a General Iterated Function System

The main theorem of this paper establishes conditions under which the "chaos game" algorithm almost surely yields the attractor of an iterated function system. The theorem holds in a very general setting, even for non contractive iterated function systems, and under weaker conditions on the random orbit of the chaos game than obtained previously.

math.MG↗

arXiv2010

The Eigenvalue Problem for Linear and Affine Iterated Function Systems

The eigenvalue problem for a linear function L centers on solving the eigen-equation Lx = rx. This paper generalizes the eigenvalue problem from a single linear function to an iterated function system F consisting of possibly an infinite number of linear or affine functions. The eigen-equation becomes F(X) = rX, where r>0 is real, X is a compact set, and F(X)is the union of f(X), for f in F. The main result is that an irreducible, linear iterated function system F has a unique eigenvalue r equal to the joint spectral radius of the functions in F and a corresponding eigenset S that is centrally symmetric, star-shaped, and full dimensional. Results of Barabanov and of Dranishnikov-Konyagin-Protasov on the joint spectral radius follow as corollaries.

math.MG↗

arXiv2008

V-Variable Fractals: Fractals with Partial Self Similarity

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 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.

math.DS↗

arXiv2005

Existence and Uniqueness of Orbital Measures

We note an elementary proof of the existence and uniqueness of a solution $% μ\in \mathbb{P(X)}$ to the equation $μ=pμ_{0}+q\hat{F}μ$. Here $\mathbb{X}$ is a topological space, $\mathbb{P(X)}$ is the set of Borel measures of unit mass on $\mathbb{X}$, $μ_{0}\in $ $\mathbb{P(X)}$ is given, $p>0$, and $q\geq 0$ with $p+q=1$. The transformation $\hat{F}:% \mathbb{P(X)\to P(X)}$ is defined by $\hat{F}\upsilon =\tsum\limits_{n=1}^{N}p_{n}\upsilon \circ f_{n}^{-1}$ where $f_{n}:\mathbb{% X\to X}$ is continuous, $p_{n}>0$ for $n=1,2,...,N$, $N$ is a finite strictly positive integer, and $\tsum\limits_{n=1}^{N}p_{n}=1$. This problem occurs in connection with iterated function systems (IFS).

math.DS↗

arXiv2003

V-variable fractals and superfractals

Deterministic and random fractals, within the framework of Iterated Function Systems, have been used to model and study a wide range of phenomena 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.

math.PR↗

arXiv2003

A Fractal Valued Random Iteration Algorithm and Fractal Hierarchy

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 key "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 $\mathbb{R}^{2}$, 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.

math.PR↗