We prove density of hyperbolicity in spaces of (i) real transcendental entire
functions, bounded on the real line, whose singular set is finite and real and
(ii) transcendental self-maps of the punctured plane which preserve the circle
and whose singular set (apart from zero and infinity) is contained in the
circle. In particular, we prove density of hyperbolicity in the famous Arnol'd
family of circle maps and its generalizations, and solve a number of other open
problems for these functions, including three conjectures by de Melo, Salomao
and Vargas.
We also prove density of (real) hyperbolicity for certain families as in (i)
but without the boundedness condition; in particular our results apply when the
function in question has only finitely many critical points and asymptotic
singularities.