Correlation functions of higher-dimensional automatic sequences

A procedure for calculating the (auto)correlation function gamma(f) (k), k is an element of Z(m), of an m-dimensional complex-valued automatic sequence f : Z(m) --> C, is presented. This is done by deriving a recursion for the vector correlation function Gamma(ker(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 Gamma(ker(f))(k), which is defined as a limit, for all k is an element of Z(m), is shown to depend only on the existence of Gamma(ker(f)) (0). This is illustrated for the higher-dimensional Thue-Morse, paper folding and Rudin-Shapiro sequences.