ON A FAMILY OF SELF-AFFINE IFS WHOSE ATTRACTORS HAVE A NON-FRACTAL TOP

Kevin G. Hare and Nikita Sidorov ()
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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

FRACTALS (fractals), 2021, vol. 29, issue 06, 1-9

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.

Keywords: Iterated Function System; Boundary (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S0218348X21501590

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