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

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 functions f : \ {0} → \ {0} that preserve the circle and whose singular set (apart from 0; ∞) is finite and contained in the circle. In particular, we prove density of hyperbolicity in the famous Arnold family of circle maps and its generalizations, and we solve a number of other open problems for these functions, including three conjectures by de Melo, Salomão, and Vargas. We also prove density of (real) hyperbolicity for certain families as in (i) but without the boundedness condition. Our results apply, in particular, when the functions in question have only finitely many critical points and asymptotic singularities, or when there are no asymptotic values and the degree of critical points is uniformly bounded. © 2015.