Gwyneth M. Stallard

Bio: Gwyneth M. Stallard is an academic researcher from Open University. The author has contributed to research in topics: Entire function & Escaping set. The author has an hindex of 23, co-authored 71 publications receiving 1532 citations.

Iteration of a class of hyperbolic meromorphic functions

Abstract: We look at the class Bn which contains those transcendental meromorphic functions f for which the finite singularities of f-n are in a bounded set and prove that, if f belongs to Bn, then there are no components of the set of normality in which fmn(z) oo as m -* oo. We then consider the class B which contains those functions f in Bl for which the forward orbits of the singularities of f-1 stay away from the Julia set and show (a) that there is a bounded set containing the finite singularities of all the functions f-n and (b) that, for points in the Julia set of f, the derivatives (fn)' have exponential-type growth. This justifies the assertion that B is a class of hyperbolic functions.

On questions of Fatou and Eremenko

Abstract: Let $f$ be a transcendental entire function and let $I(f)$ be the set of points whose iterates under $f$ tend to infinity. We show that $I(f)$ has at least one unbounded component. In the case that $f$ has a Baker wandering domain, we show that $I(f)$ is a connected unbounded set.

Fast escaping points of entire functions

Abstract: Let $f$ be a transcendental entire function and let $A(f)$ denote the set of points that escape to infinity `as fast as possible’ under iteration. By writing $A(f)$ as a countable union of closed sets, called `levels’ of $A(f)$, we obtain a new understanding of the structure of this set. For example, we show that if $U$ is a Fatou component in $A(f)$, then $\partial U\subset A(f)$ and this leads to significant new results and considerable improvements to existing results about $A(f)$. In particular, we study functions for which $A(f)$, and each of its levels, has the structure of an `infinite spider's web’. We show that there are many such functions and that they have a number of strong dynamical properties. This new structure provides an unexpected connection between a conjecture of Baker concerning the components of the Fatou set and a conjecture of Eremenko concerning the components of the escaping set.

Dynamics of meromorphic functions with direct or logarithmic singularities

Abstract: Let f be a transcendental meromorphic function and denote by J(f) the Julia set and by I(f) the escaping set. We show that if f has a direct singularity over infinity, then I(f) has an unbounded component and I(f)∩J(f) contains continua. Moreover, under this hypothesis I(f)∩J(f) has an unbounded component if and only if f has no Baker wandering domain. If f has a logarithmic singularity over infinity, then the upper box dimension of I(f)∩J(f) is 2 and the Hausdorff dimension of J(f) is strictly greater than 1. The above theorems are deduced from more general results concerning functions which have ‘direct or logarithmic tracts’, but which need not be meromorphic in the plane. These results are obtained by using a generalization of Wiman–Valiron theory. This method is also applied to complex differential equations.

Fast escaping points of entire functions

Abstract: Let $f$ be a transcendental entire function and let $A(f)$ denote the set of points that escape to infinity `as fast as possible' under iteration. By writing $A(f)$ as a countable union of closed sets, called `levels' of $A(f)$, we obtain a new understanding of the structure of this set. For example, we show that if $U$ is a Fatou component in $A(f)$, then $\partial U\subset A(f)$ and this leads to significant new results and considerable improvements to existing results about $A(f)$. In particular, we study functions for which $A(f)$, and each of its levels, has the structure of an `infinite spider's web'. We show that there are many such functions and that they have a number of strong dynamical properties. This new structure provides an unexpected connection between a conjecture of Baker concerning the components of the Fatou set and a conjecture of Eremenko concerning the components of the escaping set.

Phd by thesis

Abstract: Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Will To Appear

Abstract: This paper presents an opportunity for the uncertainty that has plagued the novel's criticism to appear as absences in the body of historical knowledge, particularly regarding the notion of life after death. Taking appearance (eg. proof of existence), as opposed to disappearance, as a universally accepted value allows this analysis to interrogate the novel's logic in relation to a variety of conventional systems whose very existence depends on the reproduction of their systems. The ineffectuality of Foucauldian disciplinary institutions in the novel establishes the threat of nonexistence. A significant relationship to Dante's Inferno is rendered, lending the appearance of language an 'enchanted' value through allusions to Dante's intentional invocation of Augustinian corporeal vision. The novel's metalanguage appears enchanted by the body of historical knowledge, particularly as the product of capitalism, discipline and Judeo-Christianity, and programmed by literary precursors William S. Burroughs, Gertrude Stein and Ernest Hemingway. Foregrounded by this complex network, an analysis of the novel’s first chapter demonstrates how an attention to appearance brings the language to life and draws the narrator, equally invested in appearance, into its realm of representation.

Dynamic rays of bounded-type entire functions

Abstract: We construct an entire function in the Eremenko-Lyubich class B whose Julia set has only bounded path-components. This answers a question of Eremenko from 1989 in the negative. On the other hand, we show that for many functions in B, in particular those of nite order, every escaping point can be connected to 1 by a curve of escaping points. This gives a partial positive answer to the aforementioned question of Eremenko, and answers a question of Fatou from 1926.