Short Separating Geodesics for Multiply Connected Domains

We consider the following questions: given a hyperbolic plane domain and a separation of its complement into two disjoint closed sets each of which contains at least two points, what is the shortest closed hyperbolic geodesic which separates these sets and is it a simple closed curve? We show that a shortest curve always exists although in general it may not be simple. However, one can also always find a shortest simple curve and we call such a geodesic a \emph{meridian} of the domain. Meridians generalize to domains of higher connectivity the notion of the equator of an annulus as the shortest geodesic which separates the complement. We show that although they are not in general uniquely defined, if one of the sets of the separation of the complement is connected, then they are unique and are also the shortest possible closed curves which separate the complement in this fashion.Comment: 20 Pages, 3 Figure

Preservation of External Rays in non-Autonomous Iteration

We consider the dynamics arising from the iteration of an arbitrary sequence of polynomials with uniformly bounded degrees and coefficients and show that, as parameters vary within a single hyperbolic component in parameter space, certain properties of the corresponding Julia sets are preserved. In particular, we show that if the sequence is hyperbolic and all the Julia sets are connected, then the whole basin at infinity moves holomorphically. This extends also to the landing points of external rays and the resultant holomorphic motion of the Julia sets coincides with that obtained earlier using grand orbits. In addition, if a finite set of external rays separate the Julia set for a particular parameter value, then the rays with the same external angles separate the Julia set for every parameter in the same hyperbolic component

On Possible Limit Functions on a Fatou Component in non-Autonomous Iteration (Research on the Theory of Random Dynamical Systems and Fractal Geometry)

The possibilities for limit functions on a Fatou component for the iteration of a single polynomial or rational function are well understood and quite restricted. In non-autonomous iteration, where one considers compositions of arbitrary polynomials with suitably bounded degrees and coefficients, one ought to observe a far greater range of behaviour. We show this is indeed the case and we exhibit a sequence of quadratic polynomials which has a bounded Fatou component on which one obtains as limit functions every member of the classical Schlicht family of suitably normalized univalent functions on the unit disc. The main idea behind this is to make use of dynamics on Siegel discs where high iterates of a single polynomial with a Siegel disc approximate the identity arbitrarily closely on compact subsets of the Siegel disc

Movement control testing of older people in community settings: description of a screening tool and intra-rater reliability

Objective: To determine the intra-rater reliability of a newly developed movement screening tool; the ‘Movement control screen for older people in community settings’. The movement screening tool aims to identify movement control impairments which can potentially influence movement function. Method: Thirty one active female recreational golfers, aged 65-77 years,carried out three movement control tests included in the screening tool. Performance was videorecorded to enable repeated ratings. Each test was evaluated by criteria which were rated as pass or fail and ratings were carried out three weeks apart to examine intra-rater reliability. Reliability was assessed using percentage agreement and Cohen’s Kappa. Results: Percentage agreementfor each test ranged from 93.0-97.3%, with an overall mean agreement of 95.5%. Kappa values for test scores ranged from 0.35-0.90. Percentage agreement for individual criteria ranged from 83.0-100.0%, with kappa values ranging from 0.00-1.00. Discussion: Acceptable intra-rater reliability was established for overall tests scores of the screening tool but certain criteria wereidentified as being less reliable than others. Recommendations are made for refinement of some criteria to improve reliability of the screening tool.<br/