Duke Mathematical Journal

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

Lasse Rempe-Gillen and Sebastian van Strien

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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 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.

Article information

Source
Duke Math. J., Volume 164, Number 6 (2015), 1079-1137.

Dates
First available in Project Euclid: 17 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1429282679

Digital Object Identifier
doi:10.1215/00127094-2885764

Mathematical Reviews number (MathSciNet)
MR3336841

Zentralblatt MATH identifier
1332.37034

Subjects
Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04]
Secondary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth) 37E10: Maps of the circle 37F15: Expanding maps; hyperbolicity; structural stability 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX]

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

Citation

Rempe-Gillen, Lasse; van Strien, Sebastian. Density of hyperbolicity for classes of real transcendental entire functions and circle maps. Duke Math. J. 164 (2015), no. 6, 1079--1137. doi:10.1215/00127094-2885764. https://projecteuclid.org/euclid.dmj/1429282679


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