# The Entropy of Cantor--like measures

Research paper by **Kathryn E. Hare, Kevin G. Hare, Brian P. M. Morris, Wanchun Shen**

Indexed on: **29 Sep '18**Published on: **29 Sep '18**Published in: **arXiv - Mathematics - Metric Geometry**

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#### 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.