×

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
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] [A] Ahlfors, L. V.: Complex Analysis, 2nd edn. New York: McGraw-Hill 1966 · Zbl 0154.31904
[2] [AM] Atiyah, M., Macdonald, I.: Introduction to Commutative Algebra. Reading: Addison-Wesley 1969 · Zbl 0175.03601
[3] [Bg] Berg, K. R.: Convolutions of invariant measures, maximal entropy. Math. Syst. Theory3, 146-150 (1969) · Zbl 0179.08301 · doi:10.1007/BF01746521
[4] [Bw] Bowen, R.: Entropy for group endomorphisms and homogeneous spaces. Trans. Am. Math. Soc.153, 401-414 (1971) · Zbl 0212.29201 · doi:10.1090/S0002-9947-1971-0274707-X
[5] [Byl] Boyd, D.: Kronecker’s theorem and Lehmer’s problem for polynomials in several variables. J. Number Theory13, 116-121 (1981) · Zbl 0447.12003 · doi:10.1016/0022-314X(81)90033-0
[6] [By2] Boyd, D.: Speculations concerning the range of Mahler’s measure. Can. Math. Bull.24, 453-469 (1981) · Zbl 0474.12005 · doi:10.4153/CMB-1981-069-5
[7] [C] Conze, J.P.: Entropie d’un groupe abélian de transformations. Z. Wahrscheinlichkeitsth. Verw. Geb.25, 11-30 (1972) · Zbl 0261.28015 · doi:10.1007/BF00533332
[8] [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
[9] [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 · doi:10.2307/2374009
[10] [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
[11] [Kt] Kato, T.: Perturbation Theory for Linear Operators. New York: Springer 1966
[12] [KS1] Kitchens, B., Schmidt, K.: Automorphisms of compact groups. Ergodic Theory Dyn. Syst.9, 691-735 (1989) · Zbl 0709.54023 · doi:10.1017/S0143385700005290
[13] [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
[14] [Lg] Lang, S.: Algebra (2nd Ed.). Reading: Addison-Wesley 1984
[15] [Lw] Lawton, W.M.: A problem of Boyd concerning geometric means of polynomials. J. Number Theory16, 356-362 (1983) · Zbl 0516.12018 · doi:10.1016/0022-314X(83)90063-X
[16] [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
[17] [Lh] Lehmer, D.H.: Factorization of cyclotomic polynomials. Ann. Math.34, 461-479 (1933) · Zbl 0007.19904 · doi:10.2307/1968172
[18] [Ln1] Lind, D.A.: Translation invariant sigma algebras on groups. Proc. Am. Math. Soc.42, 218-221 (1974) · Zbl 0276.28019 · doi:10.1090/S0002-9939-1974-0325924-X
[19] [Ln2] Lind, D.A.: Ergodie automorphisms of the infinite torus are Bernoulli. Isr. J. Math.17, 162-168 (1974) · Zbl 0284.28007 · doi:10.1007/BF02882235
[20] [Ln3] Lind, D.A.: The structure of skew products with ergodic group automorphisms. Isr. J. Math.28, 205-248 (1977) · Zbl 0365.28015 · doi:10.1007/BF02759810
[21] [Ln4] Lind, D.A.: Dynamical properties of quasihyperbolic toral automorphisms. Ergodic Theory Dyn. Syst.2, 48-68 (1982) · Zbl 0507.58034 · doi:10.1017/S0143385700009573
[22] [LW] Lind, D., Ward, T.: Automorphisms of solenoids andp-adic entropy. Ergodic Theory Dyn. Syst.8, 411-419 (1988) · Zbl 0634.22005 · doi:10.1017/S0143385700004545
[23] [Mh2] Mahler, K.: An application of Jensen’s formula to polynomials. Mathematika7, 98-100 (1960) · Zbl 0099.25003 · doi:10.1112/S0025579300001637
[24] [Mh2] Mahler, K.: On some inequalities for polynomials in several variables. J. London Math. Soc.37, 341-344 (1962) · Zbl 0105.06301 · doi:10.1112/jlms/s1-37.1.341
[25] [Mt] Matsumura, H.: Commutative Algebra. New York: Benjamin 1970 · Zbl 0211.06501
[26] [Ms] Misiurewicz, M.: A short proof of the variational principle for a ? + N on a compact space. Asterisque40, 147-157 (1975)
[27] [P] Parry, W.: Entropy and Generators in Ergodic Theory. New York: Benjamin 1969 · Zbl 0175.34001
[28] [Rh] Rohlin, V.A.: Metric properties of endomorphisms of compact commutative groups. Am. Math. Soc. Transl., Ser. 264, 244-252 (1967)
[29] [Rd] Rudin, W.: Real and Complex Analysis. New York: McGraw-Hill 1966 · Zbl 0142.01701
[30] [Sc1] Schmidt, K.: Mixing automorphisms of compact groups and a theorem by Kurt Mahler. Pac. J. Math.137, 371-385 (1989) · Zbl 0678.22002
[31] [Sc2] Schmidt, K.: Automorphisms of compact abelian groups and affine varieties. Proc. London Math. Soc., to appear
[32] [Sm1] Smyth, C.J.: A Kronecker-type theorem for complex polynomials in several variables. Can. Math. Bull.24, 447-452 (1981) · Zbl 0475.12002 · doi:10.4153/CMB-1981-068-8
[33] [Sm2] Smyth, C.J.: On measures of polynomials in several variables. Bull. Aust. Math. Soc.23, 49-63 (1981) · Zbl 0442.10034 · doi:10.1017/S0004972700006894
[34] [T1] Thomas, R.K.: The addition theorem for the entropy of transformations ofG-spaces. Trans. Am. Math. Soc.160, 119-130 (1971)
[35] [T2] Thomas, R.K.: Metric properties of transformations ofG-spaces. Trans. Am. Math. Soc.160, 103-117 (1971)
[36] [W] Walters, P.: An Introduction to Ergodic Theory. Berlin-Heidelberg-New York: Springer 1982 · Zbl 0475.28009
[37] [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 · doi:10.2307/2322543
[38] [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)
[39] [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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.