Multifractals Defined by Nonlinear Cellular Automata

Abstract

It is known that the space-time output patterns of additive cellular automata may be scaled to yield fractals. In this paper, nonadditive cellular automata rules are grouped into equivalence classes, each class defined in terms of a characteristic nonadditive rule. This rule defines a multifractal uniquely associated to each class. It is shown that these multifractals can be generated by a form of matrix substitution system, called concatenation substitution. This allows easy computation of fractal dimensions. It is likely that the multifractal spectra for each class does not possess the usual inverted U shape. A conserved quantity is found to be associated with the concatenation substitution system, and is shown to play a role similar to Langton's λ-parameter.