  Automata operating on strings of nested brackets, known as input-driven
pushdown automata, and as visibly pushdown automata, have been studied since
the 1980s. They were extended to the case of infinite strings by Alur and
Madhusudan ("Visibly pushdown languages", STOC 2004). This paper investigates
the properties of these automata under the assumption that a given infinite
string is always well-nested. This restriction enables a complete
characterization of the corresponding $\omega$-languages in terms of classical
$\omega$-regular languages and input-driven automata on finite strings. This
characterization leads to a determinization result for these automata, as well
as to the first results on their Wadge degrees.
