Summary: Univoque numbers and automatic sequences
J.-P. Allouche1
and Z.-x. Wen2
1
CNRS, LRI, UMR 8623, Universit´e Paris-Sud, B^at. 490, F-91405 Orsay Cedex,
France allouche@lri.fr
2
School of Mathematics and Statistics, Huazhong University of Science and
Technology, Wuhan 430074, Hubei, China zhi-xiong.wen@mail.hust.edu.cn
Summary. A set of binary sequences related to the iteration of unimodal contin-
uous functions of the interval [0, 1] appears in a 1982­1983 work of Cosnard and
the first author. An almost identical set of binary sequences occurs in a 1990 paper
by Erdos, Jo´o, and Komornik; it consists of expansions of 1 in univoque bases in
(1, 2) (the base is univoque if 1 admits a unique -expansion). We generalize a
result of the second author and Niu by proving, using the 1982­1983 results, that
a large class of Thue-Morse-like sequences belong to these sets of binary sequences.
The case of alphabets of size larger than 2 yields similar results.
1 Introduction
A set of binary sequences related to the iteration of continuous unimodal
functions of the interval [0, 1] was introduced at the beginning of the 80's by

Source: Allouche, Jean-Paul - Laboratoire de Recherche en Informatique, Université de Paris-Sud 11

Collections: Computer Technologies and Information Sciences