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.