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

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

2010-05-25

arxiv.org/abs/1005.4627v3

Mathematics

Dynamical Systems

41 pages, 3 figures. V3: One figure added (parameter space of the Arnol'd family); some general revision

Scientific paper

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.

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