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Binary constant-length substitutions and Mahler measures of Borwein polynomials. (English) Zbl 1461.11143

Bailey, David H. (ed.) et al., From analysis to visualization. A celebration of the life and legacy of Jonathan M. Borwein, Callaghan, Australia, September 25–29, 2017. Cham: Springer. Springer Proc. Math. Stat. 313, 303-322 (2020).
Let \(\varrho\) be a binary constant-length substitution defined on \(\{0,1\}\) as \(0 \mapsto w_0\) and \(1 \mapsto w_1\), where \(w_0,w_1\) are finite words over the alphabet \(\{0,1\}\) of equal length \(|w_0|=|w_1|=L \geq 2\). The authors show that then the extremal so-called Lyapunov exponents are given explicitly by \(\lambda_{\min}^B=0\) and \(\lambda_{\max}^B=m(p_{\varrho})\), where \(m\) stands for the logarithmic Mahler measure and \(p_{\varrho}\) is a Borwein polynomial (the one with \(-1,0,1\) coefficients), determined by \(\varrho\). In particular, if \(p_{\varrho}\) is non-reciprocal then one has \(\lambda_{\max}^B \geq m(x^3-x-1)\). As an open problem they pose a dynamical analogue of Lehmer’s problem.
For the entire collection see [Zbl 1442.00024].

MSC:

11R09 Polynomials (irreducibility, etc.)
37P99 Arithmetic and non-Archimedean dynamical systems
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