Geometry and Arithmetic of a Simple Cellular Automaton

T his pap er pr esents t he results of a study of the geomet ric and arit hmetic prope rt ies of t he graph of a simple cellular automaton, considered as a mapping of t he uni t inter val to it self. T he graph provides an examp le of a st rict ly self-similar figur e and exhibits some numeric pr op ert ies relating to Fermat numbers. In ad dit ion an interest ing density resul t is proved: t he pred ecessor set of any number in [0, 1] is dense in the interval , and t he set of kt h-orde r pr edecessors of any number is uniformly dist ributed over a par tit ion of t he interval int o 2k uniform segments.