Mahler measure and entropy for commuting automorphisms of compact groups

Abstract

We compute the joint entropy ofd commuting automorphisms of a compact metrizable group. LetR d = ℤ[u 1 ±1 ,...,[d 1 ±1 ] be the ring of Laurent polynomials ind commuting variables, andM be anR d -module. Then the dual groupX M ofM is compact, and multiplication onM by each of thed variables corresponds to an action α M of ℤ d by automorphisms ofX M . Every action of ℤ d by automorphisms of a compact abelian group arises this way. Iff∋R d , our main formula shows that the topological entropy of $$\alpha _{R_d /\left\langle f \right\rangle } $$ is given by $$h\left( {\alpha _{R_d /\left\langle f \right\rangle } } \right) = \log M\left( f \right) = \int\limits_0^1 \ldots \int\limits_0^1 {\log \left| {f\left( {e^{2\pi it_1 } , \ldots , e^{2\pi it_d } } \right)} \right|} dt_1 \ldots dt_d $$ , where M(f) is the Mahler measure off. This reduces to the classical result for toral automorphisms via Jensen's formula. While the entropy of a single automorphism of a compact group is always the logarithm of an algebraic integer, this no longer seems to hold for joint entropy of commuting automorphisms since values such as 7ζ(3)/4π2 occur. If p is a non-principal prime ideal, we show $$h\left( {\alpha _{R_d /p} } \right) = 0$$ . Using an analogue of the Yuzvinskii-Thomas addition formula, we computeh(α M ) for arbitraryR d -modulesM, and then the joint entropy for an action of ℤ d on a (not necessarily abelian) compact group. Using a result of Boyd, we characterize those α M which have completely positive entropy in terms of the prime ideals associated toM, and show this condition implies that α M is mixing of all orders. We also establish an analogue of Berg's theorem, proving that if α M has finite entropy then Haar measure is the unique measure of maximal entropy if and only if α M has completely positive entropy. Finally, we show that for expansive actions the growth rate of the number of periodic points equals the topological entropy.