We study the links between the topological complexity of an $\omega$-context free language and its degree of ambiguity. In particular, using known facts from classical descriptive set theory, we prove that non Borel $\omega$-context free languages which are recognized by Büchi pushdown automata have a maximum degree of ambiguity. This result implies that degrees of ambiguity are really not preserved by the operation $W \rightarrow W^\omega$, defined over finitary context free languages. We prove also that taking the adherence or the $\delta$-limit of a finitary language preserves neither ambiguity nor inherent ambiguity. On the other side we show that methods used in the study of $\omega$-context free languages can also be applied to study the notion of ambiguity in infinitary rational relations accepted by Büchi 2-tape automata and we get first results in that direction.
"Topology and ambiguity in ω-context free languages." Bull. Belg. Math. Soc. Simon Stevin 10 (5) 707 - 722, December 2003. https://doi.org/10.36045/bbms/1074791327