Prof.

Generalized Fractal Transforms and Self-Similar Objects in Cone Metric Spaces
H. Kunze∗ , D. La Torre∗∗ and F. Mendivil∗∗∗ , E. R. Vrscay∗∗∗∗ January 17, 2012
Department of Mathematics and Statistics, University of Guelph, Canada Department of Economics, Business and Statistics, University of Milan, Italy ∗∗∗ Department of Mathematics and Statistics, Acadia University, Canada ∗∗∗∗ Department of Applied Mathematics, University of Waterloo, Canada Abstract We use the idea of a scalarization of a cone metric to prove that the topology generated by any cone metric is equivalent to a topology generated by a related metric. We then analyze the case of an ordering cone with empty interior and we provide alternative definitions based on the notion of interior cone. Finally we discuss the implications of such cone metrics in the theory of iterated function systems and generalized fractal transforms and suggest some applications in fractal-based image analysis.
∗

∗∗

Keywords: Cone metric space, Completeness, Contractivity, Self-similarity, Digital image analysis.

1

Introduction

Properties of contractive mappings are used throughout mathematics, usually to invoke Banach’s theorem on fixed points of contractions. In the classical case of iterated function systems (IFSs), the existence of self-similar objects relies on this same theorem. Fundamental ingredients of the theory of IFS are the use of complete metric spaces and the notion of contractivity, which both depend on the definition of the underlying distance. Much recent work has focused on the extension of the notion of metric spaces and the related notion of contractivity. One such extension is the idea of cone metric spaces. In this context, the distance is no longer a positive number but a vector, in general an element of a Banach space which has been equipped with an ordering cone. In applications, it is often useful to compare two objects in multiple ways and combine these various comparisons together.