Correlation functions of higher-dimensional automatic sequences
Anna Greta Barbe, Fritz V. Haeseler · Journal of Physics A Mathematical and General · 2004
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 Γ 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 Γ ker( f ) ( k ), which is defined as a limit, for all , is shown to depend only on the existence of Γ ker( f ) (0). This is illustrated for the higher-dimensional Thue–Morse, paper folding and Rudin–Shapiro sequences.