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Expectations over Attractors of Iterated Function Systems

Borwein, Jonathan M. and Rose, Michael Expectations over Attractors of Iterated Function Systems. Applied Mathematics and Computation . (Submitted)

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    Abstract

    Motivated by the need for new mathematical tools applicable to the study of fractal point-cloud distributions, expectations of complex-valued functions defined over general `deterministic' fractal domains are considered, following the development of a measure-theoretic foundation for their analysis. In particular, we wish to understand and evaluate separation moments as defined via a normalized Borel measure supported on a self-similar subset of Rn. Previous work concerning such integrals supported over the special class of String-generated Cantor Set (SCS) fractals (see [5]) is generalised to encompass all fractal sets that can be expressed as the attractor of an Iterated Function System (IFS). The development of a generalised functional equation for expectations over IFS attractors (Proposition 3.2) enables the symbolic evaluation of certain even-order separation moments over attractors of affine IFSs, including such celebrated fractal sets as the von Koch Snowflake and Sierpinski Triangle - and more generally, any IFS attractor generated from real-world data by means of the Collage Theorem.

    Item Type: Article
    Subjects: 28-xx Measure and integration
    Faculty: UNSPECIFIED
    Depositing User: Mr. Michael Rose
    Date Deposited: 30 Sep 2015 09:53
    Last Modified: 30 Sep 2015 09:53
    URI: https://docserver.carma.newcastle.edu.au/id/eprint/1680

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