{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:51:07Z","timestamp":1760151067100,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,2,18]],"date-time":"2022-02-18T00:00:00Z","timestamp":1645142400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We study some properties of binary sequences generated by random substitutions of constant length. Specifically, assuming the alphabet {0,1}, we consider the following asymmetric substitution rule of length k: 0\u2192\u27e80,0,\u2026,0\u27e9 and 1\u2192\u27e8Y1,Y2,\u2026,Yk\u27e9, where Yi is a Bernoulli random variable with parameter p\u2208[0,1]. We obtain by recurrence the discrete probability distribution of the stochastic variable that counts the number of ones in the sequence formed after a number i of substitutions (iterations). We derive its first two statistical moments, mean and variance, and the entropy of the generated sequences as a function of the substitution length k for any successive iteration i, and characterize the values of p where the maxima of these measures occur. Finally, we obtain the parametric curves entropy-variance for each iteration and substitution length. We find two regimes of dependence between these two variables that, to our knowledge, have not been previously described. Besides, it allows to compare sequences with the same entropy but different variance and vice versa.<\/jats:p>","DOI":"10.3390\/e24020290","type":"journal-article","created":{"date-parts":[[2022,2,21]],"date-time":"2022-02-21T08:18:05Z","timestamp":1645431485000},"page":"290","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Entropy-Variance Curves of Binary Sequences Generated by Random Substitutions of Constant Length"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4708-980X","authenticated-orcid":false,"given":"Juan Carlos","family":"Nu\u00f1o","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Universidad Polit\u00e9cnica de Madrid, 28040 Madrid, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0874-6275","authenticated-orcid":false,"given":"Francisco J.","family":"Mu\u00f1oz","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Universidad Polit\u00e9cnica de Madrid, 28040 Madrid, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Fogg, N.P., Berth\u00e9, V., Ferenczi, S., Mauduit, C., and Siegel, A. (2002). Substitutions in Dynamics, Arithmetics and Combinatorics, Springer.","DOI":"10.1007\/b13861"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1109","DOI":"10.1007\/BF01026566","article-title":"On the Structure of Mandelbrot\u2019s Percolation Process and Other Random Cantor Sets","volume":"58","author":"Dekking","year":"1990","journal-title":"J. Stat. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Falconer, K. (1990). Fractal Geometry: Mathematical Foundations and Its Applications, John Wiley & Sons Ltd.","DOI":"10.2307\/2532125"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Mandelbrot, B.B. (1983). The Fractal Geometry of Nature, W. H. 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Sequence A202019 in The On-Line Encyclopedia of Integer Sequences. Available online: http:\/\/www.oeis.org."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/2\/290\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:22:16Z","timestamp":1760134936000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/2\/290"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,18]]},"references-count":17,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,2]]}},"alternative-id":["e24020290"],"URL":"https:\/\/doi.org\/10.3390\/e24020290","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,2,18]]}}}