  In this paper we study the asymptotic behaviour of two relatively new
complexity functions defined on infinite words and their relationship to
periodicity. Given a factor $u$ of an infinite word $x$, we say $u$ is closed
if it is a letter or if it is a complete first return to some factor $v$ of
$x$; otherwise $u$ is said to be open. We show that for an aperiodic word $x$
over a finite alphabet, the complexity functions that count the number of
closed and the number of open factors of $x$ of each given length are both
unbounded. More precisely, we show that if $x$ is aperiodic then the limit
inferior of the function of open complexity is infinite, and the limit superior
of the function of closed complexity is infinite on any syndetic subset of
positive integers. On the other hand, there exist aperiodic words for which
limit inferior of the closed complexity function is finite.
