Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line

Abstract

This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than 1 then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to 1. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.

A Review of Structure Engineering of Strain‐Tolerant Architectures for Stretchable Electronics

“…Second, though 4‐dimension seems to be the upper limit of dimensionality, there is an infinite number of non‐integer dimensions, as exemplified by Hausdorff or Assouad dimensions. [ 149 ] Nature is filled with such generative dimensions. Just like proteins are with primary, secondary tertiary, or even higher order structures, stretchable electronics can also form multiscale hierarchical structures with distinct mechanical or functional designs at each scale.…”