On the joint subword complexity of automatic sequences

Let the (subword) complexity of a sequence u = (u(n))(n=0)(infinity) over a finite set Sigma be the function m bar right arrow P(u)(m), where P(u)(m) denotes the number of distinct blocks u(n) ... u(n + m-1) of size m in u. In this paper, we study the complexity of (u) over right arrow (n) = (u(1)(n), ... , u(r)(n))(n=0)(infinity) when each u(i) = (u(i)(n))(n=0)(infinity), i = 1, ... , r, is a q(i)-automatic sequence over a finite set Sigma(i) and q(1), ... , q(r) >= 2 are pairwise coprime integers. As an application, we answer a question of Allouche and Shallit regarding morphic real numbers. (C) 2009 Elsevier B.V. All rights reserved.