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Combinatorial, ergodic and arithmetic properties of the Tribonacci substitution. (Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci.) (French) Zbl 1038.37010
Summary: We study combinatoric, ergodic and arithmetic properties of the fixed-point of Tribonacci substitution (first introduced by G. Rauzy) and of the related rotation of the two-dimensional torus. We give a geometric generalization of the three distances theorem and an explicit formula for the recurrence function of the fixed-point of the substitution. We state Diophantine approximation’s properties of the vector of the rotation of \(\mathbb{T}^2\): we prove that, for a suitable norm, the sequence of best approximation of this vector is the sequence of Tribonacci numbers. We compute the ergodic invariants \(F\) and \(F_C\) of the symbolic system related to the substitution.

MSC:
37B10 Symbolic dynamics
05C99 Graph theory
11B85 Automata sequences
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
05A99 Enumerative combinatorics
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