## 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:

  [A] Ahlfors, L. V.: Complex Analysis, 2nd edn. New York: McGraw-Hill 1966 · Zbl 0154.31904  [AM] Atiyah, M., Macdonald, I.: Introduction to Commutative Algebra. Reading: Addison-Wesley 1969 · Zbl 0175.03601  [Bg] Berg, K. R.: Convolutions of invariant measures, maximal entropy. Math. Syst. Theory3, 146-150 (1969) · Zbl 0179.08301  [Bw] Bowen, R.: Entropy for group endomorphisms and homogeneous spaces. Trans. Am. Math. Soc.153, 401-414 (1971) · Zbl 0212.29201  [Byl] Boyd, D.: Kronecker’s theorem and Lehmer’s problem for polynomials in several variables. J. Number Theory13, 116-121 (1981) · Zbl 0447.12003  [By2] Boyd, D.: Speculations concerning the range of Mahler’s measure. Can. Math. Bull.24, 453-469 (1981) · Zbl 0474.12005  [C] Conze, J.P.: Entropie d’un groupe abélian de transformations. Z. Wahrscheinlichkeitsth. Verw. Geb.25, 11-30 (1972) · Zbl 0261.28015  [D] Dobrowolski, E., Lawton, W., Schinzel, A.: On a problem of Lehmer. (Studies in Pure Math., pp. 133-144). Basel: Birkhäuser 1983 · Zbl 0519.12012  [E] Elsanousi, S.A.: A variational principle for the pressure of a continuous ?2 on a compact metric space. Am. J. Math.99, 77-106 (1977) · Zbl 0388.28021  [Km] Kaminski, B.: Mixing properties of two-dimensional dynamical systems with completely positive entropy. Bull. Pol. Acad. Sci., Math.27, 453-463 (1980) · Zbl 0469.28013  [Kt] Kato, T.: Perturbation Theory for Linear Operators. New York: Springer 1966  [KS1] Kitchens, B., Schmidt, K.: Automorphisms of compact groups. Ergodic Theory Dyn. Syst.9, 691-735 (1989) · Zbl 0709.54023  [KS2] Kitchens, B., Schmidt, K.: Periodic points, decidability and Markov subgroups. (Lecture Notes in Math., Vol. 1342 pp. 440-454). Berlin-Heidelberg-New York: Springer 1988 · Zbl 0664.58029  [Lg] Lang, S.: Algebra (2nd Ed.). Reading: Addison-Wesley 1984  [Lw] Lawton, W.M.: A problem of Boyd concerning geometric means of polynomials. J. Number Theory16, 356-362 (1983) · Zbl 0516.12018  [Ld] Ledrappier, F.: Un champ markovian peut être d’entropie nulle et mélangeant. C. R. Acad. Sc. Paris. Ser. A2807, 561-562 (1978) · Zbl 0387.60084  [Lh] Lehmer, D.H.: Factorization of cyclotomic polynomials. Ann. Math.34, 461-479 (1933) · Zbl 0007.19904  [Ln1] Lind, D.A.: Translation invariant sigma algebras on groups. Proc. Am. Math. Soc.42, 218-221 (1974) · Zbl 0276.28019  [Ln2] Lind, D.A.: Ergodie automorphisms of the infinite torus are Bernoulli. Isr. J. Math.17, 162-168 (1974) · Zbl 0284.28007  [Ln3] Lind, D.A.: The structure of skew products with ergodic group automorphisms. Isr. J. Math.28, 205-248 (1977) · Zbl 0365.28015  [Ln4] Lind, D.A.: Dynamical properties of quasihyperbolic toral automorphisms. Ergodic Theory Dyn. Syst.2, 48-68 (1982) · Zbl 0507.58034  [LW] Lind, D., Ward, T.: Automorphisms of solenoids andp-adic entropy. Ergodic Theory Dyn. Syst.8, 411-419 (1988) · Zbl 0634.22005  [Mh2] Mahler, K.: An application of Jensen’s formula to polynomials. Mathematika7, 98-100 (1960) · Zbl 0099.25003  [Mh2] Mahler, K.: On some inequalities for polynomials in several variables. J. London Math. Soc.37, 341-344 (1962) · Zbl 0105.06301  [Mt] Matsumura, H.: Commutative Algebra. New York: Benjamin 1970 · Zbl 0211.06501  [Ms] Misiurewicz, M.: A short proof of the variational principle for a ? + N on a compact space. Asterisque40, 147-157 (1975)  [P] Parry, W.: Entropy and Generators in Ergodic Theory. New York: Benjamin 1969 · Zbl 0175.34001  [Rh] Rohlin, V.A.: Metric properties of endomorphisms of compact commutative groups. Am. Math. Soc. Transl., Ser. 264, 244-252 (1967)  [Rd] Rudin, W.: Real and Complex Analysis. New York: McGraw-Hill 1966 · Zbl 0142.01701  [Sc1] Schmidt, K.: Mixing automorphisms of compact groups and a theorem by Kurt Mahler. Pac. J. Math.137, 371-385 (1989) · Zbl 0678.22002  [Sc2] Schmidt, K.: Automorphisms of compact abelian groups and affine varieties. Proc. London Math. Soc., to appear  [Sm1] Smyth, C.J.: A Kronecker-type theorem for complex polynomials in several variables. Can. Math. Bull.24, 447-452 (1981) · Zbl 0475.12002  [Sm2] Smyth, C.J.: On measures of polynomials in several variables. Bull. Aust. Math. Soc.23, 49-63 (1981) · Zbl 0442.10034  [T1] Thomas, R.K.: The addition theorem for the entropy of transformations ofG-spaces. Trans. Am. Math. Soc.160, 119-130 (1971)  [T2] Thomas, R.K.: Metric properties of transformations ofG-spaces. Trans. Am. Math. Soc.160, 103-117 (1971)  [W] Walters, P.: An Introduction to Ergodic Theory. Berlin-Heidelberg-New York: Springer 1982 · Zbl 0475.28009  [Yn] Young, R.M.: On Jensen’s formula and ? 0 2? log|1?e e? |d?. Am. Math. Mon.93, 44-45 (1986) · Zbl 0607.30028  [Yz1] Yuzvinskii, S.A.: Metric properties of endomorphisms of compact groups. Izv. Akad. Nauk SSSR, Ser. Math.29, 1295-1328 (1965); Engl. transl. Am. Math. Soc. Transl. (2)66, 63-98 (1968)  [Yz2] Yuzvinskii, S.A.: Computing the entropy of a group endomorphism. Sib. Mat. Z.8, 230-239 (1967) (Russian). Engl. transl. Sib. Math. J.8, 172-178 (1968)
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