An $\omega$-language is a set of infinite words over a finite alphabet $X$. We consider the class of recursive $\omega$-languages, i.e. the class of $\omega$-languages accepted by Turing machines with a Büchi acceptance condition, which is also the class $\Sigma_1^1$ of (effective) analytic subsets of $X^\omega$ for some finite alphabet $X$. We investigate the notion of ambiguity for recursive $\omega$-languages with regard to acceptance by Büchi Turing machines. We first show that the class of unambiguous recursive $\omega$-languages is the class $\Delta_1^1$ of hyperarithmetical sets. We obtain also that the $\Delta_1^1$-subsets of $X^\omega$ are the subsets of $X^\omega$ which are accepted by strictly recursive unambiguous finitely branching Büchi transition systems; this provides an effective analogue to a theorem of Arnold on Büchi transition systems [Arnold83]. Moreover, using some effective descriptive set theory, we prove that recursive $\omega$-languages satisfy the following dichotomy property. A recursive $\omega$-language $L\subseteq X^\omega$ is either unambiguous or has a great degree of ambiguity in the following sense: for every Büchi Turing machine T accepting L, there exist infinitely many $\omega$-words which have $2^{\aleph_0}$ accepting runs by T. We also show that if $L \subseteq X^\omega$ is accepted by a Büchi Turing machine T and L is an analytic but non Borel set, then the set of $\omega$-words, which have $2^{\aleph_0}$ accepting runs by T, has cardinality $2^{\aleph_0}$. In that case we say that the recursive $\omega$-language L has the maximum degree of ambiguity. We prove that it is $\Pi_2^1$-complete to determine whether a given recursive $\omega$-language is unambiguous and that it is $\Sigma_2^1$-complete to determine whether a given recursive $\omega$-language has the maximum degree of ambiguity. Moreover, using some results of set theory, we prove that it is consistent with the axiomatic system ZFC that there exists a recursive $\omega$-language in the Borel class ${\bf \Pi}^0_2$, hence of low Borel rank, which has also this maximum degree of ambiguity.