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We prove density of hyperbolicity in spaces of (i) real transcendental entirefunctions, bounded on the real line, whose singular set is finite and real and(ii) transcendental self-maps of the punctured plane which preserve the circleand whose singular set (apart from zero and infinity) is contained in thecircle. In particular, we prove density of hyperbolicity in the famous Arnol'dfamily of circle maps and its generalizations, and solve a number of other openproblems for these functions, including three conjectures by de Melo, Salom\~aoand Vargas.  We also prove density of (real) hyperbolicity for certain families as in (i)but without the boundedness condition. Our results apply, in particular, whenthe functions in question have only finitely many critical points andasymptotic singularities, or when there are no asymptotic values and the degreeof critical points is uniformly bounded.

Density of hyperbolicity for classes of real transcendental entire functions and circle maps