  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 B\"uchi pushdown automata have a maximum
degree of ambiguity. This result implies that degrees of ambiguity are really
not preserved by the operation of taking the omega power of a finitary context
free language. We prove also that taking the adherence or the delta-limit of a
finitary language preserves neither unambiguity 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 B\"uchi 2-tape automata and we get first results
in that direction.
