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arXiv · 2101.01798

On a family of Self-Affine IFS whose attractors have a non-fractal top

Kevin G. Hare·Nikita Sidorov
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Abstract

Let $0< λ< μ<1$ and $λ+μ>1$. In this note we prove that for the vast majority of such parameters the top of the attractor $A_{λ,μ}$ 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, $A_{λ, μ}$ has a 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 $λμ\ge 2^{-1/6}$, then $A_{λ,μ}$ has a non-empty interior.

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Kevin G. Hare, Nikita Sidorov. 2021-01-05. On a family of Self-Affine IFS whose attractors have a non-fractal top. https://doi.org/10.1142/s0218348x21501590

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