On the triplex substitution – combinatorial properties

Abstract

If a substitution τ over a three-letter alphabet has a positively linear complexity, that is, P τ ( n ) = C 1 n + C 2 ( n ⩾ 1 ) with C 1 , C 2 ⩾ 0 , there are only 4 possibilities: P τ ( n ) = 3 , n + 2 , 2 n + 1 or 3 n. The first three cases have been studied by many authors, but the case 3 n remained unclear. This leads us to consider the triplex substitution σ : a ↦ a b , b ↦ a c b , c ↦ a c c . Studying the factor structure of its fixed point, which is quite different from the other cases, we show that it is of complexity 3 n. We remark that the triplex substitution is also a typical example of invertible substitution over a three-letter alphabet. To cite this article: B. Tan et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).