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

Rempe-Gillen, Lasse ORCID: 0000-0001-8032-8580 and van Strien, Sebastian (2015) Density of hyperbolicity for classes of real transcendental entire functions and circle maps. Duke Mathematical Journal, 164 (6). pp. 1079-1137.

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Official URL: http://dx.doi.org/10.1215/00127094-2885764

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, Salom\~ao 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.

Item Type:	Article
Additional Information:	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
Uncontrolled Keywords:	dynamical systems, low dynamical systems, holomorphic dynamics, transcendental dynamics, density of hyperbolicity
Depositing User:	Symplectic Admin
Date Deposited:	06 Jul 2015 08:06
Last Modified:	16 Mar 2024 06:55
DOI:	10.1215/00127094-2885764
Related URLs:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/2015619