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

Abstract

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, 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.

46 pages, 3 figures. V5: Final peer-reviewed accepted manuscript, to appear in Duke Mathematical Journal. Only minor changes from the previous (significantly revised) version V4

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Cited by in corpus (4)

• Positive Transversality via transfer operators and holomorphic motions with applications to monotonicity for interval maps
• Holomorphic motions for unicritical correspondences
• The escaping set in transcendental dynamics
• Holomorphic motions, natural families of entire maps, and multiplier-like objects for wandering domains