二元常长替换与Borwein多项式的Mahler测度
Binary constant-length substitutions and Mahler measures of Borwein polynomials
摘要：
我们发现，马勒措施的每一个Borwein多项式-一个多项式的系数在$ \{-1,0,1 \}$具有非零常数项-可以表示为一个最大的李雅普诺夫指数的矩阵上循环，出现在二进制常数长度的替代谱理论。这样，Lehmer的问题高度一多项式具有最小的马勒措施成为等价于一个自然的问题，从谱理论的二进制常数长度的替代。这支持了马勒测度和动力学之间的另一种联系，超越了众所周知的马勒测度在代数动力学中作为熵的出现。
英文摘要：
We show that the Mahler measure of every Borwein polynomial -- a polynomial with coefficients in $ \{-1,0,1 \}$ having non-zero constant term -- can be expressed as a maximal Lyapunov exponent of a matrix cocycle that arises in the spectral theory of binary constant-length substitutions. In this way, Lehmer's problem for height-one polynomials having minimal Mahler measure becomes equivalent to a natural question from the spectral theory of binary constant-length substitutions. This supports another connection between Mahler measures and dynamics, beyond the well-known appearance of Mahler measures as entropies in algebraic dynamics.
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