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

TL;DR

This paper establishes the density of hyperbolic functions within specific classes of real transcendental entire functions and circle maps, solving several open problems and conjectures in the field.

Contribution

It proves the density of hyperbolicity in new classes of functions, including the Arnol'd family of circle maps, and addresses longstanding open questions and conjectures.

Findings

Density of hyperbolicity in these classes

Resolution of three conjectures by de Melo, Salom ilde{ao}, and Vargas

Applicability to functions with finitely many singularities

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.