Univoque numbers and an avatar of Thue-Morse

Univoque numbers are real numbers $\lambda >1$ such that the number 1 admits a unique expansion in base $\lambda$, i.e., a unique expansion $1 = \sum_{j \geq 0} a_j \lambda^{-(j+1)}$,with$a_j \in {0, 1, …, \lceil \lambda \rceil -1}$forevery$j \geq 0$. A variation of this definition was studied in 2002 by Komornik and Loreti, together with sequences called {\em admissible sequences}. We show how a 1983 study of the first author gives both a result of Komornik and Loreti on the smallest admissible sequence on the set ${0, 1,

…, b}$, and a result of de Vries and Komornik (2007) on the smallest univoque number belonging to the interval $(b,b+1)$, where $b$ is any positive integer. We also prove that this last number is transcendental. An avatar of the Thue-Morse sequence, namely the fixed point beginning in 3 of the morphism $3\to 31$, $2\to 30$, $1\to 03$, $0\to 02$, occurs in a “universal” manner.