Hyperbolic entire functions and the Eremenko–Lyubich class: Class B or not class B?

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 non-compactness 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 transcendental entire function 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 Eremenko–Lyubich classB of transcendental entire functions with a bounded set of singular values. We also considerably strengthen a recent characterisation of the class 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

Beyond the periodic orbit theory

The global constraints on chaotic dynamics induced by the analyticity of smooth flows are used to dispense with individual periodic orbits and derive infinite families of exact sum rules for several simple dynamical systems. The associated Fredholm determinants are of particularly simple polynomial form. The theory developed suggests an alternative to the conventional periodic orbit theory approach to determining eigenspectra of transfer operators.Comment: 29 pages Latex2

No entire function with real multipliers in class S

We prove that there is no entire transcendental function in class S with real multipliers of all repelling periodic orbits.Comment: 7 page

Further evidence supporting a role for gs signal transduction in severe malaria pathogenesis.

With the functional demonstration of a role in erythrocyte invasion by Plasmodium falciparum parasites, implications in the aetiology of common conditions that prevail in individuals of African origin, and a wealth of pharmacological knowledge, the stimulatory G protein (Gs) signal transduction pathway presents an exciting target for anti-malarial drug intervention. Having previously demonstrated a role for the G-alpha-s gene, GNAS, in severe malaria disease, we sought to identify other important components of the Gs pathway. Using meta-analysis across case-control and family trio (affected child and parental controls) studies of severe malaria from The Gambia and Malawi, we sought evidence of association in six Gs pathway candidate genes: adenosine receptor 2A (ADORA2A) and 2B (ADORA2B), beta-adrenergic receptor kinase 1 (ADRBK1), adenylyl cyclase 9 (ADCY9), G protein beta subunit 3 (GNB3), and regulator of G protein signalling 2 (RGS2). Our study amassed a total of 2278 cases and 2364 controls. Allele-based models of association were investigated in all genes, and genotype and haplotype-based models were investigated where significant allelic associations were identified. Although no significant associations were observed in the other genes, several were identified in ADORA2A. The most significant association was observed at the rs9624472 locus, where the G allele (approximately 20% frequency) appeared to confer enhanced risk to severe malaria [OR = 1.22 (1.09-1.37); P = 0.001]. Further investigation of the ADORA2A gene region is required to validate the associations identified here, and to identify and functionally characterize the responsible causal variant(s). Our results provide further evidence supporting a role of the Gs signal transduction pathway in the regulation of severe malaria, and request further exploration of this pathway in future studies

Permutable entire functions satisfying algebraic differential equations

It is shown that if two transcendental entire functions permute, and if one of them satisfies an algebraic differential equation, then so does the other one.Comment: 5 page

Boundaries of univalent Baker domains

Let $f$ be a transcendental entire function and let $U$ be a univalent Baker domain of $f$. We prove a new result about the boundary behaviour of conformal maps and use this to show that the non-escaping boundary points of $U$ form a set of harmonic measure zero with respect to $U$. This leads to a new sufficient condition for the escaping set of $f$ to be connected, and also a new general result on Eremenko's conjecture

Rigidity of escaping dynamics for transcendental entire functions

We prove an analog of Boettcher's theorem for transcendental entire functions in the Eremenko-Lyubich class B. More precisely, let f and g be entire functions with bounded sets of singular values and suppose that f and g belong to the same parameter space (i.e., are *quasiconformally equivalent* in the sense of Eremenko and Lyubich). Then f and g are conjugate when restricted to the set of points which remain in some sufficiently small neighborhood of infinity under iteration. Furthermore, this conjugacy extends to a quasiconformal self-map of the plane. We also prove that this conjugacy is essentially unique. In particular, we show that an Eremenko-Lyubich class function f has no invariant line fields on its escaping set. Finally, we show that any two hyperbolic Eremenko-Lyubich class functions f and g which belong to the same parameter space are conjugate on their sets of escaping points.Comment: 28 pages; 2 figures. Final version (October 2008). Various modificiations were made, including the introduction of Proposition 3.6, which was not formally stated previously, and the inclusion of a new figure. No major changes otherwis

Fatou flowers and parabolic curves

In this survey we collect the main results known up to now (July 2015) regarding possible generalizations to several complex variables of the classical Leau-Fatou flower theorem about holomorphic parabolic dynamics