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On A Family Of Self-Affine Ifs Whose Attractors Have A Non-Fractal Top

Author

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  • KEVIN G. HARE

    (Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1, Canada)

  • NIKITA SIDOROV

    (Department of Mathematics, The University of Manchester, Manchester M13 9PL, UK)

Abstract

Let 0 < λ < μ < 1 and λ + μ > 1. In this paper, we prove that for the vast majority of such parameters the top of the planar attractor Aλ,μ of the IFS {(λx,μy), (μx + 1 − μ,λy + 1 − λ)} is the graph of a continuous, strictly increasing function. Despite this, for most parameters, Aλ,μ has a lower box dimension strictly greater than 1, showing that the upper boundary is not representative of the complexity of the fractal. Finally, we prove that if λμ ≥ 2−1/6, then Aλ,μ has a non-empty interior.

Suggested Citation

  • Kevin G. Hare & Nikita Sidorov, 2021. "On A Family Of Self-Affine Ifs Whose Attractors Have A Non-Fractal Top," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(06), pages 1-9, September.
    • Handle: RePEc:wsi:fracta:v:29:y:2021:i:06:n:s0218348x21501590
    DOI: 10.1142/S0218348X21501590
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    Keywords

    Iterated Function System; Boundary;

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