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arXiv · 1810.00201

The Entropy of Cantor--like measures

Kathryn E. Hare·Kevin G. Hare·Brian P. M. Morris·Wanchun Shen
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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 $2\leq d 0$, $\sum p_{i}=1$. 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.

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Kathryn E. Hare, Kevin G. Hare, Brian P. M. Morris, Wanchun Shen. 2018-09-29. The Entropy of Cantor--like measures. https://arxiv.org/abs/1810.00201

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