Hyperbolic entire functions and the EremenkoLyubich class: Class $\mathcal{B}$ or not class $\mathcal{B}$?
Abstract
Hyperbolicity plays an important role in the study of dynamical systems, and is a key concept in the iteration of rational functions of one complex variable. Hyperbolic systems have also been considered in the study of transcendental entire functions. There does not appear to be an agreed definition of the concept in this context, due to complications arising from the noncompactness of the phase space. In this article, we consider a natural definition of hyperbolicity that requires expanding properties on the preimage of a punctured neighbourhood of the isolated singularity. We show that this definition is equivalent to another commonly used one: a {\tef} is hyperbolic if and only if its postsingular set is a compact subset of the Fatou set. This leads us to propose that this notion should be used as the general definition of hyperbolicity in the context of entire functions, and, in particular, that speaking about hyperbolicity makes sense only within the \emph{EremenkoLyubich class} $\mathcal{B}$ of transcendental entire functions with a bounded set of singular values. We also considerably strengthen a recent characterisation of the class $\mathcal{B}$, by showing that functions outside of this class cannot be expanding with respect to a metric whose density decays at most polynomially. In particular, this implies that no transcendental entire function can be expanding with respect to the spherical metric. Finally we give a characterisation of an analogous class of functions analytic in a hyperbolic domain.
 Publication:

arXiv eprints
 Pub Date:
 February 2015
 arXiv:
 arXiv:1502.00492
 Bibcode:
 2015arXiv150200492R
 Keywords:

 Mathematics  Complex Variables;
 Mathematics  Dynamical Systems;
 37F10;
 30D05;
 30D15;
 30D20;
 37F15
 EPrint:
 20 pages, 2 figures. V2: Final accepted manuscript, to appear in Math. Zeitschrift. Various minor corrections from V1