A NONCOMMUTATIVE DIFFERENTIAL GEOMETRIC METHOD TO FRACTAL GEOMETRY (I) (REPRESENTATIONS OF CUNTZ ALGEBRAS OF HAUSDORFF TYPE ON SELF SIMILAR FRACTAL SETS)

In a series of papers with the same title we shall give a noncommutative differential geometric method to fractal geometry and show a possiblity of a description of phase transtions in statistical mechanics in terms of fractal geometry. In this paper we consider the representations of Cuntz algebras on proper/improper self similar fractal sets. We can obtain the following results:

1. We give a sufficient condition for the existence of the representations of Cuntz algebras on self similar fractal sets.

2. The Cuntz algebra has representations on proper self similar fractal sets, which are called Hausdorff representations and the criterion whether they are equivalent or not can be given.

3. We can construct representations on improper self similar fractal sets which are called of Hausdorff type and show that they are not equivalent to that on a proper self similar fractal set.

This is an expository note and the full will be given in the forthcoming paper(⁶).