Glasnik Matematicki, Vol. 38, No.2 (2003), 377-393.

APPROXIMATE RESOLUTIONS AND THE FRACTAL CATEGORY

Takahisa Miyata and Tadashi Watanabe

Department of Mathematics and Information Sciences, Faculty of Education, Yamaguchi University, Yamaguchi-City, 753-8513, Japan
e-mail: tadashi@po.yb.cc.yamaguchi-u.ac.jp

Abstract.   This paper concerns the theory of approximate resolutions and its application to fractal geometry. In this paper, we first characterize a surjective map f : X → Y between compact metric spaces in terms of a property on any approximate map f : X → Y where p : X → X and q : Y → Y are any choices of approximate resolutions of X and Y, respectively. Using this characterization, we construct a category whose objects are approximate sequences so that the box-counting dimension, which was defined for approximate resolutions by the authors, is invariant in this category. To define the morphisms of the category, we introduce an equivalence relation on approximate maps and define the morphisms as the equivalence classes.

2000 Mathematics Subject Classification.   54C56, 28A80.

Key words and phrases.   Approximate resolution, surjective map, box-counting dimension, category.