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Acta Mathematica Hungarica

, Volume 159, Issue 2, pp 563–588 | Cite as

The entropy of Cantor-like measures

  • K. E. Hare
  • K. G. HareEmail author
  • B. P. M. Morris
  • W. Shen
Article
  • 27 Downloads

Abstract

By a Cantor-like measure we mean the unique self-similar probability measure \(\mu\) satisfying \(\mu = \sum^{m-1}_{i=0} p_{i}{\mu} {\circ} S^{-1}_{i}\) where \(S_{i}(x) = \frac{x}{d} + \frac{i}{d} \cdot \frac{d-1}{m-1}\) for integers \(2 \leq d < m \leq 2d - 1\) and probabilities \(p_{i} > 0, {\sum}p_{i} = 1\). In the uniform case \((p_{i} = 1/m {\rm for all} i)\) 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.

Key words and phrases

entropy Cantor measure Hausdorff dimension 

Mathematics Subject Classification

28A78 28A80 28D20 

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  • K. E. Hare
    • 1
  • K. G. Hare
    • 1
    Email author
  • B. P. M. Morris
    • 2
  • W. Shen
    • 1
  1. 1.Dept. of Pure MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.Dept. of MathematicsStanford UniversityStanfordUSA

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