IJPAM: Volume 24, No. 2 (2005)

PERIODICITY THEOREMS FOR GRADED FRACTAL
BUNDLES RELATED TO CLIFFORD STRUCTURES

Julian \Lawrynowicz$^1$, Osamu Suzuki$^2$
$^1$Institute of Physics
University of \Lódz
Pomorska 149/153, PL-90-236 \Lódz, POLAND
and
$^1$Institute of Mathematics
Polish Academy of Sciences
\Lódz Branch, Banacha 22, PL-90-238 \Lódz, POLAND
e-mail: jlawryno@uni.lodz.pl
$^2$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 $A_{1}^{1},A_{2}^{1},...,A_{2p-1}^{1}$ of a Clifford algebra $Cl_{2p-1}(\mathbb{C})$, $p=2,3,...,$ we consider the sequence

\begin{displaymath} \begin{split} A_{\alpha}^{q+1}= \left( \begin{array}{cc} A_{... ...iI_{p,q}\\ -iI_{p,q}&0 \end{array} \right) \end{split}\tag{1} \end{displaymath} ()

of generators of Clifford algebras $Cl_{2p+2q-1}(\mathbb{C}),\ q=1, 2, ...,$ and the sequence of the corresponding systems of closed squares $Q_{q+1}^{\alpha}$ of diameter 1, centred at the origin of $\mathbb{C}$, where $I_{p,q}=I_{2^{p+q-2}}$. Then we decompose each $Q_{q}^{\alpha}$ into the corresponding $4^{p+q-2}$ equal squares with sides parallel to the sides of the original square. In the case of
\begin{displaymath} A_{\alpha}^{q}=\left( a _{\alpha j}^{qk} \right),\quad j,k=1,2,...,2^{p+q-2},\tag{2} \end{displaymath} ()

we include to the object constructed all closed squares corresponding to the matrix elements equal $\alpha _{\alpha j}^{qk}$ whenever it is different from zero. We may say that we consider the bundle $(\Sigma_{\alpha})$ of $a _{\alpha j}^{qk}$-graded fractals
\begin{displaymath} \begin{split} &\Sigma _{\alpha}= \left( Q_{q}^{\alpha} \righ... ...+2, ...\ {\mbox {\rm for}}\ \alpha \geq 2p, \end{split}\tag{3} \end{displaymath} ()

where [ ] denotes the function ``entier"; we endow them with the functions
\begin{displaymath} \begin{split} &g_{q}^{\alpha}\left( a_{\alpha j}^{qk}; z \ri... ...a}\left( z \right) \neq a_{\alpha j}^{qk}, \end{split}\tag{4} \end{displaymath} ()

where $g_{q}^{\alpha}$ is the gradating function: $g_{q}^{\alpha}\left(z \right)=a_{\alpha j}^{qk}$ inside the square corresponding to the pair $(j,k)$. By (3), for $\alpha \geq 2p$ each $Q_{q}^{\alpha}$ is decomposed into $4^{p+q-1}$ equal squares. We obtain periodicity theorems for the sequences of the gradating functions $g_{q}^{\alpha}(z)$. 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