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The entropy of Cantor-like measures

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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.

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Authors and Affiliations

  1. Dept. of Pure Mathematics, University of Waterloo, Waterloo, Ont., N2L 3G1, Canada

    K. E. Hare, K. G. Hare & W. Shen

  2. Dept. of Mathematics, Stanford University, Stanford, CA, 94305, USA

    B. P. M. Morris

Authors
  1. K. E. Hare
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  2. K. G. Hare
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  3. B. P. M. Morris
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  4. W. Shen
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Corresponding author

Correspondence to K. G. Hare.

Additional information

Research of K. E. Hare was supported by NSERC Grant 2016-03719.

Research of K. G. Hare was supported by NSERC Grant 2014-03154.

Research of B. Morris was supported by Fields Undergraduate Summer Research Program.

Research of W. Shen was supported by the NSERC USRA program and Grants 2014-03154 and 2016-03719.

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Hare, K., Hare, K., Morris, B. et al. The entropy of Cantor-like measures. Acta Math. Hungar. 159, 563–588 (2019). https://doi.org/10.1007/s10474-019-00962-1

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  • DOI: https://doi.org/10.1007/s10474-019-00962-1

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