Substitution invariant Beatty sequences

Substitution Invariant Beatty Sequences

with θ irrational and taken to satisfy 0 < θ < 1; plainly this may be assumed without loss of generality. Evidently (fn) is a sequence of zeros and ones. Denote by w0 and w1 words on the alphabet {0, 1} ; that is, finite strings in the letters 0 and 1. Then the sequence (fn) is said to be invariant under the substitution W given by W : 0 −→ w0, 1 −→ w1, if the infinite strings fθ = f1f2f3 . . ....

Invariant games and non-homogeneous Beatty sequences

We characterize all the pairs of complementary non-homogenous Beatty sequences (An) n≥0 and (Bn) n≥0 for which there exists an invariant game having exactly {(An, Bn) | n ≥ 0} ∪ {(Bn, An) | n ≥ 0} as set of P-positions. Using the notion of Sturmian word and tools arising in symbolic dynamics and combinatorics on words, this characterization can be translated to a decision procedure relying only...

Nonhomogeneous Beatty Sequences Leading to Invariant Games

We characterize pairs of complementary non-homogeneous Beatty sequences (An)n>0 and (Bn)n>0, with the restriction A1 = 1 and B1 ≥ 3, for which there exists an invariant take-away game having {(An, Bn), (Bn, An) | n > 0} ∪ {(0, 0)} as set of P -positions. Using the notion of Sturmian word arising in combinatorics on words, this characterization can be translated into a decision procedure relying...

Substitution invariant cutting sequences

Complete Characterization of Substitution Invariant Sturmian Sequences

We provide a complete characterization of substitution invariant inhomogeneous bidirectional pointed Sturmian sequences. The result is analogous to that obtained by Berthé et al. [5] and Yasutomi [21] for one-directional Sturmian words. The proof is constructive, based on the geometric representation of Sturmian words by a cut-and-project scheme.