Topological measures, image transformations and self-similar sets

Abstract

This paper introduces a novel idea: the concept of an image transformation. We also introduce the closely related concept of a quasihomomorphism, and study the properties of these mathematical objects, and give several examples. In particular we investigate iterated systems of image transformations, which we believe give a more realistic approach to the study of so called self-similar structures in nature than what is obtained by iterated function systems.

The homology of spaces of simple topological measures

“…Image transformations were introduced in [4] as natural continuous topological measure preserving transformations. Hence they are natural tools to consider when calculating the various homology groups of spaces of simple topological measures.…”

“…Image transformations were introduced in [4] as natural topological measure preserving transformations. We include their definition and construction from solid sets:…”

“…The image transformation Ψ : X → X * defined by Ψ (A) = A * where A * = {µ ∈ X * : µ(A) = 1} was presented in [4]. It is noteworthy that this image transformation is, through the correspondence in Theorem 21, analogous to the * -operation of Definition 17.…”

“…Let q : A(X) → A(X) be an image transformation. The adjoint map q * : X * → X * given by (q * µ)(A) = µ(q(A)) for A ∈ A(X) and µ ∈ Q(X) was shown in [4] to be continuous. By Corollary 25(iii), q * must have a fixed point in X * .…”

“…A comprehensive treatment of the adjoint operation may be found in [4]. The proof of Proposition 44 goes through unchanged with X * and Y * replaced by Q(X) and Q(Y ), but we do not need the more general result here.…”