We study the complexity of the infinite word uβ associated with the
Rényi expansion of 1 in an irrational base β > 1.
When β is the golden ratio, this is the well known Fibonacci word,
which is Sturmian, and of complexity C(n) = n + 1.
For β such that
dβ(1) = t1t2...tm is finite we provide a simple description of
the structure of special factors of the word uβ. When tm=1
we show that
C(n) = (m - 1)n + 1. In the cases when t1 = t2 = ... tm-1or
t1 > max{t2,...,tm-1} we show that the first difference
of the complexity function C(n + 1) - C(n ) takes value in
{m - 1,m} for every n, and consequently we determine the
complexity of uβ. We show that
uβ is an Arnoux-Rauzy sequence if and only if
dβ(1) = tt...t1. On the example of
β = 1 + 2cos(2π/7), solution of X3 = 2X2 + X - 1, we illustrate
that the structure of special factors is more complicated for
dβ(1) infinite eventually periodic.
The complexity for this word is equal to 2n+1.