Periodic domains of quasiregular maps

We consider the iteration of quasiregular maps of transcendental type from Rd to Rd. We give a bound on the rate at which the iterates of such a map can escape to infinity in a periodic component of the quasi-Fatou set. We give examples which show that this result is best possible. Under an additional hypothesis, which is satisfied by all uniformly quasiregular maps, this bound can be improved to be the same as those in a Baker domain of a transcendental entire function. We construct a quasiregular map of transcendental type from R3 to R3 with a periodic domain in which all iterates tend locally uniformly to infinity. This is the first example of such behaviour in a dimension greater than two. Our construction uses a general result regarding the extension of biLipschitz maps. In addition, we show that there is a quasiregular map of transcendental type from R3 to R3 which is equal to the identity map in a half-space

The size and topology of quasi-Fatou components of quasiregular maps

We consider the iteration of quasiregular maps of transcendental type from Rd to Rd. In particular we study quasi-Fatou components, which are defined as the connected components of the complement of the Julia set. Many authors have studied the components of the Fatou set of a transcendental entire function, and our goal in this paper is to generalise some of these results to quasi-Fatou components. First, we study the number of com- plementary components of quasi-Fatou components, generalising, and slightly strengthening, a result of Kisaka and Shishikura. Second, we study the size of quasi-Fatou components that are bounded and have a bounded complementary component. We obtain results analogous to those of Zheng, and of Bergweiler, Rippon and Stallard. These are obtained using techniques which may be of interest even in the case of transcendental entire functions

The provision of education and training for healthcare professionals through the medium of the internet

This paper describes a new initiative to provide Internet based courses to student and professional occupational therapists in four centres in the UK, Belgium the Netherlands and Sweden. The basis of this collaborative Occupational Therapy Internet School (OTIS) is the concept of the “Virtual College”. This comprises the design and implementation of a sophisticated Internet-based system through which courses can be managed, prepared and delivered online in an effective fashion, and where students can communicate both with the staff and their peers. The aim is to support and facilitate the whole range of educational activities within a remote electronic environment. A major feature of the course organisation is the adoption of a problem-based approach in which students will collaborate internationally to propose effective intervention in given case study scenarios. The paper outlines the rationale for OTIS, the content and structure of the courseware, the technical specification of the system and evaluation criteria. In addition to the more conventional web-based learning facilities generally offered, a number of agent-based approaches are being adopted to assist in the management of the course by ensuring the proper delivery of course materials and to assist the functioning of project groups. </p

Healthy ageing and home: The perspectives of very old people in five European countries

This paper reports on in-depth research, using a grounded theory approach, to examine the ways in which very old people perceive healthy ageing in the context of living alone at home within urban settings in five European countries. This qualitative study was part of a cross-national project entitled ENABLE-AGE which examined the relationship between home and healthy ageing. Interviews explored the notion of healthy ageing, the meaning and importance of home, conceptualisations of independence and autonomy and links between healthy ageing and home. Data analysis identified five ways in which older people constructed healthy ageing: home and keeping active; managing lifestyles, health and illness; balancing social life; and balancing material and financial circumstances. Older people reflected on their everyday lives at home in terms of being engaged in purposeful, meaningful action and evaluated healthy ageing in relation to the symbolic and practical affordances of the home, contextualised within constructions of their national context. The research suggests that older people perceive healthy ageing as an active achievement, created through individual, personal effort and supported through social ties despite the health, financial and social decline associated with growing older. The physicality and spatiality of home provided the context for establishing and evaluating the notion of healthy ageing, whilst the experienced relationship between home, life history and identity created a meaningful space within which healthy ageing was negotiated

The bungee set in quasiregular dynamics

In complex dynamics, the bungee set is defined as the set points whose orbit is neither bounded nor tends to infinity. In this paper we study, for the first time, the bungee set of a quasiregular map of transcendental type. We show that this set is infinite, and shares many properties with the bungee set of a transcendental entire function. By way of contrast, we give examples of novel properties of this set in the quasiregular setting. In particular, we give an example of a quasiconformal map of the plane with a non-empty bungee set; this behaviour is impossible for an analytic homeomorphism

Which sequences are orbits?

In the study of discrete dynamical systems, we typically start with a function from a space into itself, and ask questions about the properties of sequences of iterates of the function. In this paper we reverse the direction of this study. In particular, restricting to the complex plane, we start with a sequence of complex numbers and study the functions (if any) for which this sequence is an orbit under iteration. This gives rise to questions of existence and of uniqueness. We resolve some questions, and show that these issues can be quite delicate

The dynamics of quasiregular maps of punctured space

The Fatou-Julia iteration theory of rational and transcendental entire functions has recently been extended to quasiregular maps in more than two real dimensions. Our goal in this paper is similar; we extend the iteration theory of analytic self-maps of the punctured plane to quasiregular self-maps of punctured space. We define the Julia set as the set of points for which the complement of the forward orbit of any neighbourhood of the point is a finite set. We show that the Julia set is non-empty, and shares many properties with the classical Julia set of an analytic function. These properties are stronger than those known to hold for the Julia set of a general quasiregular map of space. We define the quasi-Fatou set as the complement of the Julia set, and generalise a result of Baker concerning the topological properties of the components of this set. A key tool in the proof of these results is a version of the fast escaping set. We generalise various results of Marti-Pete concerning this set, for example showing that the Julia set is equal to the boundary of the fast escaping set

Children’s experiences of domestic violence and abuse: siblings’ accounts of relational coping

This article explores how young people see their relationships, particularly their sibling relationships, in families affected by domestic violence, and how relationality emerges in their accounts as a resource to build an agentic sense of self. The ‘voice’ of children is largely absent from domestic violence literature, which typically portrays them as passive, damaged and relationally incompetent. Children’s own understandings of their relational worlds are often overlooked, and consequently existing models of children’s social interactions give inadequate accounts of their meaning-making-in-context. Drawn from a larger study of children’s experiences of domestic violence and abuse, this paper uses two case studies of sibling relationships to explore young people’s use of relational resources, for coping with violence in the home. The paper explores how relationality and coping intertwine in young people’s accounts, and disrupts the taken for granted assumption that children’s ‘premature caring’ or ‘parentification’ is (only) pathological in children’s responses to domestic violence. This has implications for understanding young people’s experiences in the present, and supporting their capacity for relationship building in the future