# IJPAM: Volume 24, No. 2 (2005)

**PERIODICITY THEOREMS FOR GRADED FRACTAL**

BUNDLES RELATED TO CLIFFORD STRUCTURES

BUNDLES RELATED TO CLIFFORD STRUCTURES

Julian awrynowicz, Osamu Suzuki

Institute of Physics

University of ódz

Pomorska 149/153, PL-90-236 ódz, POLAND

and

Institute of Mathematics

Polish Academy of Sciences

ódz Branch, Banacha 22, PL-90-238 ódz, POLAND

e-mail: jlawryno@uni.lodz.pl

Department of Computer and System Analysis

College of Humanities and Sciences

Nihon University

Sakurajosui 3-25-40, Setagaya-ku, Tokyo, 156-8550, JAPAN

e-mail: osuzuki@am.chs.nihon-u.ac.jp

Institute of Physics

University of ódz

Pomorska 149/153, PL-90-236 ódz, POLAND

and

Institute of Mathematics

Polish Academy of Sciences

ódz Branch, Banacha 22, PL-90-238 ódz, POLAND

e-mail: jlawryno@uni.lodz.pl

Department of Computer and System Analysis

College of Humanities and Sciences

Nihon University

Sakurajosui 3-25-40, Setagaya-ku, Tokyo, 156-8550, JAPAN

e-mail: osuzuki@am.chs.nihon-u.ac.jp

**Abstract.**Given generators
of a
Clifford algebra
, we consider the sequence

() |

() |

*bundle*

*of*-

*graded fractals*

() |

() |

*gradating function*: inside the square corresponding to the pair . By (3), for each is decomposed into equal squares. We obtain

*periodicity theorems*for the sequences of the gradating functions . They play a crucial role in the further theory and applications to dynamical systems on infinite-dimensional Clifford algebras, analysis of a complex variable (value distribution theory, cluster sets, prime ends, Picard's Theorems), and physical systems, especially alloys (binary, ternary, etc).

**Received: **June 27, 2005

**AMS Subject Classification: **81R25, 32L25, 53A50, 15A66

**Key Words and Phrases: **Clifford algebra, bilinear form, quadratic form

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2005

**Volume:** 24

**Issue:** 2