Derived sequences of complementary symmetric Rote sequences

Abstract

Complementary symmetric Rote sequences are binary sequences which have factor complexityC(n) = 2nfor all integersn≥ 1 and whose languages are closed under the exchange of letters. These sequences are intimately linked to Sturmian sequences. Using this connection we investigate the return words and the derived sequences to the prefixes of any complementary symmetric Rote sequencevwhich is associated with a standard Sturmian sequenceu. We show that any non-empty prefix ofvhas three return words. We prove that any derived sequence ofvis coding of three interval exchange transformation and we determine the parameters of this transformation. We also prove thatvis primitive substitutive if and only ifuis primitive substitutive. Moreover, if the sequenceuis a fixed point of a primitive morphism, then all derived sequences ofvare also fixed by primitive morphisms. In that case we provide an algorithm for finding these fixing morphisms.