## 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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