Summary
We compute the joint entropy ofd commuting automorphisms of a compact metrizable group. LetR d = ℤ[u ±11 ,...,[d ±11 ] 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 ℤd by automorphisms ofX M . Every action of ℤd 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
, 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 ℤd 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.
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References
[A] Ahlfors, L. V.: Complex Analysis, 2nd edn. New York: McGraw-Hill 1966
[AM] Atiyah, M., Macdonald, I.: Introduction to Commutative Algebra. Reading: Addison-Wesley 1969
[Bg] Berg, K. R.: Convolutions of invariant measures, maximal entropy. Math. Syst. Theory3, 146–150 (1969)
[Bw] Bowen, R.: Entropy for group endomorphisms and homogeneous spaces. Trans. Am. Math. Soc.153, 401–414 (1971)
[Byl] Boyd, D.: Kronecker's theorem and Lehmer's problem for polynomials in several variables. J. Number Theory13, 116–121 (1981)
[By2] Boyd, D.: Speculations concerning the range of Mahler's measure. Can. Math. Bull.24, 453–469 (1981)
[C] Conze, J.P.: Entropie d'un groupe abélian de transformations. Z. Wahrscheinlichkeitsth. Verw. Geb.25, 11–30 (1972)
[D] Dobrowolski, E., Lawton, W., Schinzel, A.: On a problem of Lehmer. (Studies in Pure Math., pp. 133–144). Basel: Birkhäuser 1983
[E] Elsanousi, S.A.: A variational principle for the pressure of a continuous ℤ2 on a compact metric space. Am. J. Math.99, 77–106 (1977)
[Km] Kaminski, B.: Mixing properties of two-dimensional dynamical systems with completely positive entropy. Bull. Pol. Acad. Sci., Math.27, 453–463 (1980)
[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)
[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
[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)
[Ld] Ledrappier, F.: Un champ markovian peut être d'entropie nulle et mélangeant. C. R. Acad. Sc. Paris. Ser. A2807, 561–562 (1978)
[Lh] Lehmer, D.H.: Factorization of cyclotomic polynomials. Ann. Math.34, 461–479 (1933)
[Ln1] Lind, D.A.: Translation invariant sigma algebras on groups. Proc. Am. Math. Soc.42, 218–221 (1974)
[Ln2] Lind, D.A.: Ergodie automorphisms of the infinite torus are Bernoulli. Isr. J. Math.17, 162–168 (1974)
[Ln3] Lind, D.A.: The structure of skew products with ergodic group automorphisms. Isr. J. Math.28, 205–248 (1977)
[Ln4] Lind, D.A.: Dynamical properties of quasihyperbolic toral automorphisms. Ergodic Theory Dyn. Syst.2, 48–68 (1982)
[LW] Lind, D., Ward, T.: Automorphisms of solenoids andp-adic entropy. Ergodic Theory Dyn. Syst.8, 411–419 (1988)
[Mh2] Mahler, K.: An application of Jensen's formula to polynomials. Mathematika7, 98–100 (1960)
[Mh2] Mahler, K.: On some inequalities for polynomials in several variables. J. London Math. Soc.37, 341–344 (1962)
[Mt] Matsumura, H.: Commutative Algebra. New York: Benjamin 1970
[Ms] Misiurewicz, M.: A short proof of the variational principle for a ℤ N+ on a compact space. Asterisque40, 147–157 (1975)
[P] Parry, W.: Entropy and Generators in Ergodic Theory. New York: Benjamin 1969
[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
[Sc1] Schmidt, K.: Mixing automorphisms of compact groups and a theorem by Kurt Mahler. Pac. J. Math.137, 371–385 (1989)
[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)
[Sm2] Smyth, C.J.: On measures of polynomials in several variables. Bull. Aust. Math. Soc.23, 49–63 (1981)
[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
[Yn] Young, R.M.: On Jensen's formula and ∫ 2π0 log|1−e eθ|dθ. Am. Math. Mon.93, 44–45 (1986)
[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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The authors gratefully acknowledge support from NSF Grant DMS-8706284, the IBM Thomas J. Watson Research Center, the Milliman Endowment, and SERC Award B85318868
Oblatum 24-VI-1989 & 30-XI-1989
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Lind, D., Schmidt, K. & Ward, T. Mahler measure and entropy for commuting automorphisms of compact groups. Invent Math 101, 593–629 (1990). https://doi.org/10.1007/BF01231517
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DOI: https://doi.org/10.1007/BF01231517