A note on univoque self-Sturmian numbers

Abstract

We compare two sets of (infinite) binary sequences whose suffixes satisfy extremal conditions: one occurs when studying iterations of a unimodal continuous map from the unit interval into itself, but it also characterizes univoque real numbers; the other is an equivalent definition of characteristic Sturmian sequences. As a corollary to our study we obtain that a real number β in (1, 2) is univoque and self-Sturmian if and only if the β-expansion of 1 is of the form 1v, where v is a characteristic Sturmian sequence beginning itself in 1.

Markoff–Lagrange spectrum of one‐sided shifts

“…The smallest complexity for an aperiodic sequence is 𝑝 𝐚 (𝑛) = 𝑛+1, which is attained precisely by Sturmian sequences; see, for example, [18,Theorem 2.1.5]. By [10, Proposition 2.1], all aperiodic words with lim sup 𝑝 𝐚 (𝑛) 𝑛 < 4 3 are essentially equal to 𝝈(1 ∞ ) with 𝝈 = (𝜏 𝑗 𝑛 ,𝑘 𝑛 ) 𝑛⩾1 ∈ 𝑆 ∞ (and 𝑘 𝑛 ⩽ 2𝑗 𝑛 +1 or (𝑗 𝑛 , 𝑘 𝑛 ) = (0, 2)). Without conditions on 𝑗 𝑛 , 𝑘 𝑛 , we get the following upper bound for 𝑝 𝝈(1 ∞ ) (𝑛), which is optimal since…”

“…For example, the limit word of the sequence $$(\tau _{0,1})^\infty$$ is the Fibonacci word. Note that Sturmian sequences that are $$\sup$$‐words are also considered in [4].…”

The natural extensions of β-transformations which generalize baker's transformations

“…The connection between β-shifts and mechanical words has been studied and has disclosed it's fertility [2,3,8,[11][12][13]. Proposition 2.1 indicates that this interdisciplinary study inevitably involves lexicographic order between mechanical words.…”

Extremal properties of (epi)Sturmian sequences and distribution modulo $1$

“…Thus x ≥ y, and hence ∀k ≥ 0, 0x ≤ T k (s) ≤ 1y ≤ 1x. Now suppose that s has the property that there exists a binary sequence u such that (5) ∀k ≥ 0, 0u ≤ T k (s) ≤ 1u.…”

“…In [5] JPA uses Theorem 1 to prove that a Sturmian sequence s on {0, 1} belongs to the set Γ (see ( 13)) if and only if there exists a characteristic Sturmian sequence u beginning with 1 such that s = 1u. (In particular, a Sturmian sequence belonging to Γ must begin with 11.)…”