Typical orbits of quadratic polynomials with a neutral fixed point: Brjuno type

We describe the topological behavior of typical orbits of complex quadratic polynomials P_alpha(z)=e^{2\pi i alpha} z+z^2, with alpha of high return type. Here we prove that for such Brjuno values of alpha the closure of the critical orbit, which is the measure theoretic attractor of the map, has zero area. Then combining with Part I of this work, we show that the limit set of the orbit of a typical point in the Julia set is equal to the closure of the critical orbit.Comment: 38 pages, 5 figures; fixed the issues with processing the figure

Preventive measures for fire-related injuries and their risk factors in residential buildings: a systematic review

Abstract: Background: Every year, a large number of people lose their lives or become injured seriously as a result of fires. Fires in buildings pose a great threat to resident safety. The aim of this systematic review is to identify preventive measures for fire-related injuries in residential buildings, taking into account associated risk factors. Methods: In this study, a systematic review was performed of all studies conducted in the field of residential building fires, influencing factors and available safety procedures. From the earliest record up to 7 July 2017, databases of PubMed, Web of Science/Knowledge, and Scopus were searched and selected articles included in the study. Results: A total of 5,613 published articles were examined, of which 30 were finally found to meet the inclusion criteria. The findings of the study were included in two main groups of preventive measures and risk factors for residential building fires and related injuries. Regarding preventive measures, the factors to reduce the risk of fire-related injuries raised in the studies under review included rule amendments, changes and modification of the environment, behavior change such as emergency evacuation during fire occurrence, improvements to emergency medical services, and awareness-raising. Also, many of the studies showed that areas with a large number of young children, older people, people with physical and mental disabilities, alcohol and drug addicts, smokers, single-family households and low-income families were particularly at risk of fire-related injuries and deaths. Conclusions: There are features in residential buildings and attributes among residents that can be related to fire hazard and fire-related injuries and deaths. The most important point of this study is to focus on preventive strategies including environmental modification, promotion of safety rules and changes in risk behavior among residents. Policy makers should pay more attention to these important issues in order to promote safety and injury prevention in relation to building fires

Estimates on Fatou coordinates, and dynamics near irrationally indifferent fixed points

We present an optimal estimate on the Fatou coordinates that plays a key role in the study of the dynamics of holomorphic maps near irrationally indifferent fixed points. The estimates have been established using quasi-confrormal techniques, and are a major improvement over known estimates obtained using classical methods of complex analysis. Time permitting, we discuss some of the applications of the new estimates.Non UBCUnreviewedAuthor affiliation: Imperial College LondonPostdoctora

Typical orbits of quadratic polynomials with a neutral fixed point: Non-Brjuno type

We investigate the quantitative and analytic aspects of the near-parabolic renormalization scheme introduced by Inou and Shishikura in 2006. These provide techniques to study the dynamics of some holomorphic maps of the form f .z/ D e 2 i˛z C O.z2 /, including the quadratic polynomials e 2 i˛z C z 2 , for some irrational values of ˛. The main results of the paper concern finescale features of the measure-theoretic attractors of these maps, and their dependence on the data. As a bi-product, we establish an optimal upper bound on the size of the maximal linearization domain in terms of the Siegel-Brjuno-Yoccoz series of ˛

Analytic maps of parabolic and elliptic type with trivial centralisers

We prove that for a dense set of irrational numbers α\alphaα, the analytic centraliser of the map e2πiαz+z2e^{2\pi i \alpha} z+ z^2e2πiαz+z2 near 000 is trivial. We also prove that some analytic circle diffeomorphisms in the Arnol’d family, with irrational rotation numbers, have trivial centralisers. These provide the first examples of such maps with trivial centralisers