Correlation functions of higher-dimensional automatic sequences

Abstract

A procedure for calculating the (auto)correlation function , of an m-dimensional complex-valued automatic sequence , is presented. This is done by deriving a recursion for the vector correlation function Γker(f)(k) whose components are the (cross)correlation functions between all sequences in the finite set ker(f), the so-called kernel of f which contains all properly defined decimations of f. The existence of Γker(f)(k), which is defined as a limit, for all , is shown to depend only on the existence of Γker(f)(0). This is illustrated for the higher-dimensional Thue–Morse, paper folding and Rudin–Shapiro sequences.