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Mahler measure and entropy for commuting automorphisms of compact groups. (English) Zbl 0774.22002

In this important paper, the authors study actions of \(\mathbb Z^ d\) by automorphisms of compact abelian groups. If \(R_ d=\mathbb Z[u_ 1^{\pm1},\dots, u_ d^{\pm 1}]\) is the ring of Laurent polynomials in \(d\) commuting variables and if \(M\) is an \(R_ d\) module, then the dual group \(X_ M\) of \(M\) is compact, and multiplication on \(M\) by each of the \(d\) variables corresponds to an action \(\alpha_ M\) of \(\mathbb Z^ d\) by automorphisms of \(X_ M\). Every action of \(\mathbb Z\) by automorphisms of the compact abelian group arises in this way. The main point of the paper is a formula for \(h(\alpha_ M)\), the topological entropy of \(\alpha_ M\). By a series of algebraic arguments, this reduces to computing \(h(\alpha_ M)\) in the special case \(M=R_ d/\langle f\rangle\), where \(f\in R_ d\). Their surprising result is that, for such \(M\), \[ h(\alpha_ M)=\log\mathbf M(f)=\int_ 0^ 1 \dots \int_ 0^ 1 \log| f(e^{2\pi it_ 1}, \dots,e^{2\pi it_ d})|\, dt_ 1\dots dt_ d. \] Here \(\mathbf M(f)\) is the Mahler measure of \(f\), originally introduced by Mahler to prove inequalities for polynomials for use in transcendence theory. In the case \(d=1\), if \(f(x)=a\prod_ \xi (x-\xi)\), Jensen’s formula shows that \(\log\mathbf M(f)=\log | a|+\sum_ \xi \max(0,\log| \xi|)\). In this form, one recognizes a familiar formula for the entropy of a toral automorphism. It is satisfying to see \(\log\mathbf M(f)\) appearing as a dynamically meaningful quantity in the general case considered here.
The proof of this formula and its extension to arbitrary \(\mathbb Z^ d\)-actions occupies the first four sections of the paper. Section 5 contains a number of instructive examples. In section 6, these results are used to characterize the modules \(M\) for which \(\alpha_ M\) has entropy 0, and those for which \(\alpha_ M\) has completely positive entropy. In the final section 7, it is shown that, for expansive actions, the growth rate of the number of periodic points equals the topological entropy.

MSC:

22D40 Ergodic theory on groups
28D20 Entropy and other invariants
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
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References:

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