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15 April 2015 Density of hyperbolicity for classes of real transcendental entire functions and circle maps
Lasse Rempe-Gillen, Sebastian van Strien
Duke Math. J. 164(6): 1079-1137 (15 April 2015). DOI: 10.1215/00127094-2885764


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:C{0}C{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.


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Lasse Rempe-Gillen. Sebastian van Strien. "Density of hyperbolicity for classes of real transcendental entire functions and circle maps." Duke Math. J. 164 (6) 1079 - 1137, 15 April 2015.


Published: 15 April 2015
First available in Project Euclid: 17 April 2015

zbMATH: 1332.37034
MathSciNet: MR3336841
Digital Object Identifier: 10.1215/00127094-2885764

Primary: 37F10
Secondary: 30D05 , 37E05 , 37E10 , 37F15

Keywords: density of hyperbolicity , dynamical systems , holomorphic dynamics , low dynamical systems , transcendental dynamics

Rights: Copyright © 2015 Duke University Press


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Vol.164 • No. 6 • 15 April 2015
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