Linear Symmetric Cellular Automata Provide Salem's Singular Function

• Published in: Real analysis exchange Vol. 50; no. 1; p. 53
• Main Author: Kawaharada, Akane
• Format: Journal Article
• Language: English
• Published: East Lansing Michigan State University Press 01.01.2025
• Subjects: Cellular automata; Classification; Dimensional analysis; Fractal analysis; Fractal geometry; Fractals; Mathematical functions; Numerical analysis; Parameters; Symmetry
• ISSN: 0147-1937; 1930-1219
• DOI: 10.14321/realanalexch.1707241119

Salem's singular function is strictly increasing, continuous, and has a derivative equal to zero almost everywhere in [ 0 , 1 ] ; it is also known as de Rham's singular function or Lebesgue's singular function. The parameter of Salem's singular function Lα is α ∈ ( 0 , 1 ) and α ≠ 1 / 2 . Our previous studies have shown that for some cases of which the limit set of spatio-temporal pattern of a cellular automaton (CA) is fractal, Salem's singular function with α = 1 / 3 , 1 / 4 , or 1 / 5 is given by projecting the pattern onto the time axis. However, except for the above examples, it was unclear whether or not there exists a CA that gives Salem's singular function such that the parameter α takes other values. In this paper, we construct linear symmetric CAs giving L1/(2D+1) and L1/(2D+1) for each dimension D ⩾ 1 . This implies that there exist linear symmetric CAs that give Salem's function with a parameter α equal to the multiplicative inverse of any odd integer greater than or equal to 3 . The objective of this study is twofold. The first objective is to classify fractals from an applied mathematical perspective. Since it is difficult and costly to visualize high-dimensional patterns as they are, it is expected to become a new means of high-dimensional fractal analysis by using a function that preserves more information about the original fractal, rather than a fractal dimension that characterizes the pattern with a single numerical value. The second objective is to characterize and classify `pathological functions', which are functions with deviant, irregular, or counterintuitive properties. By considering functions for various CAs, it is expected to be possible to characterize and classify pathological functions associated with fractals.