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

The iterated minimum modulus and conjectures of Baker and Eremenko

In transcendental dynamics significant progress has been made by studying points whose iterates escape to infinity at least as fast as iterates of the maximum modulus. Here we take the novel approach of studying points whose iterates escape at least as fast as iterates of the minimum modulus, and obtain new results related to Eremenko's conjecture and Baker's conjecture, and the rate of escape in Baker domains. To do this we prove a result of wider interest concerning the existence of points that escape to infinity under the iteration of a positive continuous function

Slow escaping points of quasiregular mappings

This article concerns the iteration of quasiregular mappings on Rd and entire functions on C. It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate. Moreover, an asymptotic rate of escape result is proved that is new even for transcendental entire functions. Let f:Rd→Rd be quasiregular of transcendental type. Using novel methods of proof, we generalise results of Rippon and Stallard in complex dynamics to show that the Julia set of f contains points at which the iterates fn tend to infinity arbitrarily slowly. We also prove that, for any large R, there is a point x with modulus approximately R such that the growth of |fn(x)| is asymptotic to the iterated maximum modulus Mn(R,f)

Entire functions with Julia sets of positive measure

Let f be a transcendental entire function for which the set of critical and asymptotic values is bounded. The Denjoy-Carleman-Ahlfors theorem implies that if the set of all z for which |f(z)|>R has N components for some R>0, then the order of f is at least N/2. More precisely, we have log log M(r,f) > (N/2) log r - O(1), where M(r,f) denotes the maximum modulus of f. We show that if f does not grow much faster than this, then the escaping set and the Julia set of f have positive Lebesgue measure. However, as soon as the order of f exceeds N/2, this need not be true. The proof requires a sharpened form of an estimate of Tsuji related to the Denjoy-Carleman-Ahlfors theorem.Comment: 17 page

Escape rate and Hausdorff measure for entire functions

The escaping set of an entire function is the set of points that tend to infinity under iteration. We consider subsets of the escaping set defined in terms of escape rates and obtain upper and lower bounds for the Hausdorff measure of these sets with respect to certain gauge functions.Comment: 24 pages; some errors corrected, proof of Theorem 2 shortene

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

Historic landscape character and sense of place

This is an Author's Accepted Manuscript of an article published in Landscape Research, 2013, Vol. 38, Issue 2 pp.179-202, copyright Taylor & Francis, available online at: http://www.tandfonline.com/10.1080/01426397.2012.672642.Most studies of landscape character within archaeology and historical geography have focused on morphological features such as whether settlement patterns were nucleated or dispersed, but this paper discusses how adding depth to this, for example by studying place-names, vernacular architecture, and the territorial structures within which a landscape was managed in the past, gives us a far greater understanding of its texture and meaning to local communities. In two case-studies in southern Essex, for example, it is shown how the connections that once existed between inland and coastal communities can be used today to promote public access to the countryside. A further case study, in southwest England, shows how field-/place-names and vernacular architecture also make an important contribution to our appreciation of the time depth and complexity of landscape character.Arts and Humanities Research Council (AHRC)Southend-on-Sea Borough Counci

The impact of relationship quality on life satisfaction and well-being in dementia caregiving dyads: findings from the IDEAL study

Objectives: The quality of the relationship between people with dementia and their informal caregiver maybe an important determinant of life satisfaction and well-being for both members of the dyad. Taking a dyadic perspective, the aim of this study was to examine whether self- and partner-rated relationship quality influences life satisfaction and well-being for both people with dementia and their caregivers. Design and methods: Using data from 1283 dyads in the Improving the Experience of Dementia and Enhancing Active Life (IDEAL) cohort, we examined the impact of current relationship quality on life satisfaction and well-being in dementia caregiving dyads. Data were analysed using the Actor–Partner Interdependence Model (APIM) framework. Results: Self-rated relationship quality was associated with own life satisfaction and well-being for both people with dementia and caregivers. Partner-rated relationship quality did not influence own life satisfaction or well-being for either member of the dyad. Conclusion: This study is the first to use the APIM framework to explore the dyadic associations between relationship quality and life satisfaction and well-being in a large cohort of dementia caregiving dyads. The obtained findings suggest that the individual perception of the quality of the caregiving relationship held by each member of the caregiving dyad is an important factor for that member’s life satisfaction and well-being, while the partner’s perception of relationship quality is not. The findings highlight the importance of considering the individual perspective of both the person with dementia and the caregiver and enabling each to maintain positive perceptions of relationship quality

Correction to: Emergence of knock-down resistance in the Anopheles gambiae complex in the Upper River Region, The Gambia, and its relationship with malaria infection in children.

Unfortunately, the original article [1] contained an error mistakenly carried forward by the Production department handling this article whereby some figures and their captions were interchanged. The correct figures (Figs. 1, 2, 3, 4, 5) and captions are presented in this erratum. The original article has also been updated to reflect this correction