Abstract
We deal with outstanding properties of the Thue–Morse sequence and consider some of its appearances in combinatorial, symbolic, and physical problems. In particular, we consider the solution of a problem in semigroups theory, and additionally, we consider a system of difference equations associated with a transmission of waves problem, studied in Avishai and Berend, Phys Rev B 45:6873–688, 1991, [1], and Avishai and Berend, Phys Rev B 45:2717–2774, 1992 [2]. Such system has interesting properties from the dynamical point of view, particularly what concerns to periodic points and asymptotic behaviors of most of the non-periodic orbits. Additionally, we state similar problems using Fibonacci and Rudin–Shapiro sequences.
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