Hankel determinants of a Sturmian sequence

<abstract><p>Let $ \tau $ be the substitution $ 1\to 101 $ and $ 0\to 1 $ on the alphabet $ \{0, 1\} $. The fixed point of $ \tau $ obtained starting from 1, denoted by $ {\bf{s}} $, is a Sturmian sequence. We first give a characterization of $ {\bf{s}} $ using $ f $-representation. Then we show that the distribution of zeros in the determinants induces a partition of integer lattices in the first quadrant. Combining those properties, we give the explicit values of the Hankel determinants $ H_{m, n} $ of $ {\bf{s}} $ for all $ m\ge 0 $ and $ n\ge 1 $.</p></abstract>

Cobham’s Theorem and Automaticity

We make certain bounds in Krebs’ proof of Cobham’s theorem explicit and obtain corresponding upper bounds on the length of a common prefix of an aperiodic [Formula: see text]-automatic sequence and an aperiodic [Formula: see text]-automatic sequence, where [Formula: see text] and [Formula: see text] are multiplicatively independent. We also show that an automatic sequence cannot have arbitrarily large factors in common with a Sturmian sequence.

Derived sequences of complementary symmetric Rote sequences

Complementary symmetric Rote sequences are binary sequences which have factor complexity C(n) = 2n for all integers n ≥ 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 sequence v which is associated with a standard Sturmian sequence u. We show that any non-empty prefix of v has three return words. We prove that any derived sequence of v is coding of three interval exchange transformation and we determine the parameters of this transformation. We also prove that v is primitive substitutive if and only if u is primitive substitutive. Moreover, if the sequence u is a fixed point of a primitive morphism, then all derived sequences of v are also fixed by primitive morphisms. In that case we provide an algorithm for finding these fixing morphisms.

Une propriété de domination convexe pour les orbites sturmiennes

Let x = (x0; x1; … ) be a N-periodic sequence of integers (N ≥ 1), and s a sturmian sequence with the same barycenter (and also N-periodic, consequently). It is shown that, for affine functions ∝: RN(N) → R which are increasing relatively to some order ≤2 on RN(N) (the space of all N-periodic sequences), the average of |∝| on the orbit of x is greater than its average on the orbit of s.

Sturmian Sequences and Invertible Substitutions

Combinatorics International audience It is known that a Sturmian sequence S can be defined as a coding of the orbit of rho (called the intercept of S) under a rotation of irrational angle alpha (called the slope). On the other hand, a fixed point of an invertible substitution is Sturmian. Naturally, there are two interrelated questions: (1) Given an invertible substitution, we know that its fixed point is Sturmian. What is the slope and intercept? (2) Which kind of Sturmian sequences can be fixed by certain non-trivial invertible substitutions? In this paper we give a unified treatment to the two questions. We remark that though the results are known, our proof is very elementary and concise.