  The Arnoux-Rauzy-Poincar\'e multidimensional continued fraction algorithm is
obtained by combining the Arnoux-Rauzy and Poincar\'e algorithms. It is a
generalized Euclidean algorithm. Its three-dimensional linear version consists
in subtracting the sum of the two smallest entries to the largest if possible
(Arnoux-Rauzy step), and otherwise, in subtracting the smallest entry to the
median and the median to the largest (the Poincar\'e step), and by performing
when possible Arnoux-Rauzy steps in priority. After renormalization it provides
a piecewise fractional map of the standard $2$-simplex. We study here the
factor complexity of its associated symbolic dynamical system, defined as an
$S$-adic system. It is made of infinite words generated by the composition of
sequences of finitely many substitutions, together with some restrictions
concerning the allowed sequences of substitutions expressed in terms of a
regular language. Here, the substitutions are provided by the matrices of the
linear version of the algorithm. We give an upper bound for the linear growth
of the factor complexity. We then deduce the convergence of the associated
algorithm by unique ergodicity.
