Open and closed complexity of infinite words

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.