Mathematics Colloquium

Lasse Rempe

(Liverpool)

"Density of hyperbolicity in families of real transcendental entire functions."


Date: Mon, May 11, 2009
Time: 17:15
Place: Research II Lecture Hall

Abstract: One-dimensional real and complex dynamics studies the iteration of functions of one real or complex variable. In this context, the so-called hyperbolic maps are of particular interest as dynamical systems dominated by the simplest possible type of behaviour: convergence to a stable periodic cycle. It is a central question in the field whether such maps are dense within (suitable) parameter spaces. For the real quadratic family, Lyubich, and independently Graczyk and Swiatek, gave a positive answer in celebrated work of the 1990s. More recently, Kozlovski, Shen and van Strien were able to prove density of hyperbolicity in the space of real polynomials of a given degree. (In the complex setting, the problem remains open even for the simplest parameter spaces.) In the talk I will discuss joint work with Sebastian van Strien that establishes density of hyperbolicity for many parameter spaces of real transcendental entire functions. To our knowledge, this is the first result of this kind. The proof uses recent structural results on the dynamical behavior of transcendental entire functions near infinity (R., to appear in Acta Math.)