The Entropy of Cantor--like measures

TL;DR

This paper investigates the entropy and Hausdorff dimension of Cantor-like measures, providing methods for precise computation in the uniform case and bounds in the non-uniform case.

Contribution

It introduces a way to compute the entropy and Hausdorff dimension of Cantor-like measures, especially in the uniform case, and establishes bounds for the non-uniform case.

Findings

Exact entropy and Hausdorff dimension can be computed for uniform measures.

Bounds on entropy are established for non-uniform measures.

The methods allow arbitrary precision calculations.

Abstract

By a Cantor-like measure we mean the unique self-similar probability measure $\mu $ satisfying $\mu =\sum_{i=0}^{m-1}p_{i}\mu \circ S_{i}^{-1}$ where $% S_{i}(x)=\frac{x}{d}+\frac{i}{d}\cdot \frac{d-1}{m-1}$ for integers $2\leq d<m\le 2d-1$ and probabilities $p_{i}>0$, $\sum p_{i}=1$. In the uniform case ($p_{i}=1/m$ for all $i$) we show how one can compute the entropy and Hausdorff dimension to arbitrary precision. In the non-uniform case we find bounds on the entropy.