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

Abstract

Let $0<\lambda <\mu <1$ and $\lambda +\mu >1$. In this note we prove that for the vast majority of such parameters the top of the attractor $A_{\lambda ,\mu }$ of the IFS $\{(\lambda x,\mu y),(\mu x+1-\mu ,\lambda y+1-\lambda )\}$ is the graph of a continuous, strictly increasing function. Despite this, for most parameters, $A_{\lambda ,\mu }$ 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 $\lambda \mu \ge 2^{-1/6}$, then $A_{\lambda ,\mu }$ has a non-empty interior.

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References in corpus (2)

• On a family of self-affine sets: topology, uniqueness, simultaneous expansions
• Two-dimensional self-affine sets with interior points, and the set of uniqueness