Matrix maps for substitution sequences in the biquaternion representation

A biquaternion representation for any 2×2 matrix substitution sequence is constructed. In this representation, a 2×2 matrix is equivalent to a biquaternion which can be resolved into a scalar part and a vector part, representing the trace map (scalar map) and the generalized antitrace map (vector map) of the matrix, respectively. Then a geometrical description for the recursion process is developed. This approach is attested to be a powerful tool for studying one-dimensional aperiodic structures. The minimum dimension of the generalized antitrace map in a general case is shown to be exactly nine. The applications of the biquaternion representation in one-dimensional tight-binding electronic systems are explored. The commutativity between cluster matrices is introduced for seeking extended states in the periodic-doubling sequence. The cyclicity of matrix maps for the Fibonacci sequence is restudied, and more critical wave functions and long-range scaling properties in this system can be found. The geometric structure of the Thue-Morse sequence is examined by presenting an invariant symmetry axis and two invariant planes in its vector map.