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A renormalisation analysis of the one-dimensional Thue-Morse aperiodic chain

The one-dimensional Thue-Morse sequence with a generating matrix of determinant 0 has been suggested to be intermediate between a periodic structure and a standard quasi-periodic structure like the Fibonacci sequence. The renormalisation procedure developed in the authors' previous study of the one-dimensional Fibonacci quasi-crystal is applied here to discuss the electronic spectrum of a tight-binding Thue-Morse aperiodic chain. The resulting integrated electronic density of states indeed shows a structure that is more like that of a periodic chain than a Fibonacci quasi-crystal. They suggest that the analytical procedure can be extended to study one-dimensional quasi-crystals of any other kind, the two-dimensional Penrose lattice as well as three-dimensional real quasi-crystals.