$|v|$ is the *period* of $\uu$. If $\uu$ is not eventually periodic, it is *aperiodic*. A factor $w$ of $\uu$ is a *right special* factor if there exist at least two letters $a,b \in \mathcal{A}$ such that $wa, wb$ belong to the language $\mathcal{L}(\uu)$. A *left special* factor is defined analogously.

An infinite word $\uu$ is eventually periodic if and only if $\mathcal{L}(\uu)$ contains only finitely many right special factors. Equivalently, $\uu$ is eventually periodic if and only if its factor complexity $\mathcal{C}_{\uu}$ is bounded. On the other hand, the factor complexity of any aperiodic word satisfies $\mathcal{C}_{\uu}(n)\geq n+1$ for every $n \in \mathbb{N}$.

An infinite word $\uu$ with $ \mathcal{C}_{\uu}(n)= n+1$ for each $n \in \mathbb{N}$ is called Sturmian. A Sturmian word is *standard* (or *characteristic*) if each of its prefixes is a left special factor.

Derivated words {#SecDerivatedWords}
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Consider a prefix $w$ of an infinite recurrent word $\uu$. Let $i<j$ be two consecutive occurrences of $w$ in $\uu$. The string $u_iu_{i+1}\cdots u_{j-1}$ is a *return word* to $w$ in $\uu$. The set of all return words to $w$ in $\uu$ is denoted by $\mathcal{R}_{\uu}(w)$. Let us suppose that the set of return words to $w$ is finite, 