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Dynamically defined fractals with overlapping constructions


Since the seventies there has been a growing interest in different branches of the sciences (including physics, chemistry, biology, finance, etc.) about a new phenomenon called "fractals''. As a result of this, since the mid-eighties mathematicians have been investigating the structure of fractal sets. The simplest fractals are the so called self similar ones. These are fractals which are unions of their own similar copies (I.e. scaled down versions of the original). These similar copies (so called cylinders) may be pair wise disjoint. This is the easy case, which has been well understood since mid eighties. However, there may be significant overlapping between the cylinders. This latter case is much more problematic to analyse. The first result in hand ling this more difficult case was due to Falconer 1988. Inspired by that work, M. Pollicott and K. Simon (the host and the researcher of this project, respectively) worked out a method which is nowadays called applying the "transversality conditions. In f act, this is the only known method to give a good understanding of a typical member of a one parameter family of fractals of overlapping construction.. The aim of our project is to develop new and more effective tools to handle fractals arising from such overlapping constructions. Such fractals naturally occur in many important physical models. To achieve this aim we investigate four major cases: The Hausdorff dimension of the fat Sierpinski triangle, The Fourier transform of self-similar measures for an overlapping construction, IFS which are contracting on average, Fractals of overlapping construction used in fractal image recognition. These problems are explained in detail in the "Project objectives''. To solve any of these specific problems will require the invention of new methods. These new techniques can then subsequently be applied in the general case to give a better understanding of all fractals arising from overlapping constructions.

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Gibbet Hill Road
United Kingdom

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