Lipschitz functions and approximate resolutions

Abstract

A Lipschitz function between metric spaces is an important notion in fractal geometry as it is well known to have a close connection to fractal dimension. On the other hand, the theory of approximate resolutions has been developed by Mardešić and Watanabe. In this theory maps f :X→Y between general spaces are represented by approximate maps f : X→ Y between approximate systems for any approximate resolutions p :X→ X and q :Y→ Y , and the approximate maps f give useful information about the properties of the maps f. In this paper, we describe a new method of using the theory of approximate resolutions to study Lipschitz functions. More precisely, first of all, given a Hausdorff space X and a normal sequence U with a reasonable condition, a new metric d U which induces the given topology is defined, and Lipschitz functions with respect to the metrics induced by normal sequences are characterized by a property of the normal sequences. Secondly, using this metric, for each compact metric space X and for each approximate resolution p :X→ X of X with a reasonable condition, a new metric d p which is topologically equivalent to the given metric is defined, and the properties of those metrics are investigated. Lipschitz functions between continua with the metrics induced by approximate resolutions are characterized by approximate resolutions. As an application, contraction maps are characterized, and a sufficient condition in terms of approximate resolutions for the existence of a unique fixed point is obtained.