Summary: Univoque numbers and automatic sequences
and Z.-x. Wen2
CNRS, LRI, UMR 8623, Universit´e Paris-Sud, B^at. 490, F-91405 Orsay Cedex,
School of Mathematics and Statistics, Huazhong University of Science and
Technology, Wuhan 430074, Hubei, China email@example.com
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.
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