The Entropy of Cantor--like measures

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 for integers $2\le d<m\le 2d-1$ 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.

21 pages, 7 figures