Box-counting dimension in one-dimensional random geometry of multiplicative cascades

TL;DR

This paper studies the box-counting dimension of images of sets under random multiplicative cascade functions, revealing that it depends intricately on the set structure and differs from Hausdorff dimension results.

Contribution

It provides an explicit formula for the box-counting dimension of images of convergent sequences under random cascades and establishes bounds for general sets.

Findings

Explicit formula for box-counting dimension of convergent sequences.

Box-counting dimension depends on set structure, not just dimensions.

Bounds for box-counting dimension of images of general sets.

Abstract

We investigate the box-counting dimension of the image of a set $E \subset \mathbb{R}$ under a random multiplicative cascade function $f$. The corresponding result for Hausdorff dimension was established by Benjamini and Schramm in the context of random geometry, and for sufficiently regular sets, the same formula holds for the box-counting dimension. However, we show that this is far from true in general, and we compute explicitly a formula of a very different nature that gives the almost sure box-counting dimension of the random image $f(E)$ when the set $E$ comprises a convergent sequence. In particular, the box-counting dimension of $f(E)$ depends more subtly on $E$ than just on its dimensions. We also obtain lower and upper bounds for the box-counting dimension of the random images for general sets $E$.