DOI: 10.25537/dm.2019.SB-57-77

After Furstenberg had provided a first glimpse of remarkable rigidity phenomena associated with the joint action of several commuting automorphisms (or endomorphisms) of a compact Abelian group, further key examples motivated the development of an extensive theory of such actions. \par Two of Mahler's achievements, the recognition of the significance of Mahler measure of multivariate polynomials in relating the lengths and heights of products of polynomials in terms of the corresponding quantities for the constituent factors, and his work on additive relations in fields, have unexpectedly played important roles in the study of entropy and higher order mixing for these actions. \par This article briefly surveys these connections between Mahler's work and dynamics. It also sketches some of the dynamical outgrowths of his work that are very active today, including the investigation of the Fuglede-Kadison determinant of a convolution operator in a group von Neumann algebra as a noncommutative generalisation of Mahler measure, as well as Diophantine questions related to the growth rates of periodic points and their relation to entropy.

11-03, 37A44, 37P15, 11K60, 11R06

- [M31]. K. Mahler, Eine arithmetische Eigenschaft der Taylor-Koeffizienten rationaler Funktionen, Proc. Akad. Wet. Amsterdam 38 (1935), 50-60.
- [M143]. K. Mahler, An application of Jensen's formula to polynomials, Mathematika 7 (1960), 98-100.
- 1. V. I. Arnol'd and A. Avez, Ergodic problems of classical mechanics, Benjamin, New York, 1968.
- 2. L. P. Bowen, A measure-conjugacy invariant for free group actions, Ann. of Math. (2) 171 (2010), no. 2, 1387-1400.
- 3. D. W. Boyd, Mahler's measure for polynomials in several variables, This publication.
- 4. Ch. Deninger, Fuglede-Kadison determinants and entropy for actions of discrete amenable groups, J. Amer. Math. Soc. 19 (2006), no. 3, 737-758.
- 5. H. Derksen and D. Masser, Linear equations over multiplicative groups, recurrences, and mixing I, Proc. Lond. Math. Soc. (3) 104 (2012), no. 5, 1045-1083.
- 6. H. Derksen and D. Masser, Linear equations over multiplicative groups, recurrences, and mixing II, Indag. Math. (N.S.) 26 (2015), no. 1, 113-136.
- 7. V. Dimitrov, Convergence to the Mahler measure and the distribution of periodic points for algebraic noetherian \(\mathbb{Z}^d\)-actions, preprint, arXiv:1611.04664v2 [math.DS].
- 8. V. Dimitrov, Diophantine approximations by special points and applications to dynamics and geometry, PhD Thesis, Yale University (2017).
- 9. F. J. Dyson and H. Falk, Period of a discrete cat mapping, Amer. Math. Monthly 99 (1992), no. 7, 603-614.
- 10. B. Fuglede and R. V. Kadison, Determinant theory in finite factors, Ann. of Math. (2) 55 (1952), 520-530.
- 11. H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation, Math. Systems Theory 1 (1967), 1-49.
- 12. A. O. Gel'fond, Transcendental and algebraic numbers, Dover, New York, 1960.
- 13. P. Habegger, Diophantine approximations on definable sets, Selecta Math. (N.S.) 24 (2018), no. 2, 1633-1675.
- 14. P. R. Halmos, On automorphisms of compact groups, Bull. Amer. Math. Soc. 49 (1943), 619-624.
- 15. P. R. Halmos, Lectures on ergodic theory, Chelsea, New York, 1960.
- 16. B. Hayes, Fuglede-Kadison determinants and sofic entropy, Geom. Funct. Anal. 26 (2016), no. 2, 520-606.
- 17. D. Kerr and H. Li, Ergodic theory: Independence and dichotomies, Springer, Cham, 2016.
- 18. B. Kitchens and K. Schmidt, Automorphisms of compact groups, Ergodic Theory Dynam. Systems 9 (1989), no. 4, 691-735.
- 19. B. Kitchens and K. Schmidt, Mixing sets and relative entropies for higher-dimensional Markov shifts, Ergodic Theory Dynam. Systems 13 (1993), no. 4, 705-735.
- 20. W. M. Lawton, A problem of Boyd concerning geometric means of polynomials, J. Number Theory 16 (1983), no. 3, 356-362.
- 21. F. Ledrappier, Un champ markovien peut être d'entropie nulle et mélangeant, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 7, A561-A563 (French, with English summary).
- 22. H. Li and A. Thom, Entropy, determinants, and \(l^2\)-torsion, J. Amer. Math. Soc. 27 (2014), no. 1, 239-292.
- 23. D. A. Lind, Ergodic automorphisms of the infinite torus are bernoulli, Israel J. Math. 17 (1974), 162-168.
- 24. D. A. Lind and K. Schmidt, A survey of algebraic actions of the discrete Heisenberg group, Russian Math. Surveys 70 (2015), no. 4, 657-714.
- 25. D. A. Lind, K. Schmidt, and E. Verbitskiy, Entropy and growth rate of periodic points of algebraic \(\mathbb{Z}^d\)-actions, Contemp. Math. 532 (2010), 195-211.
- 26. D. A. Lind, K. Schmidt, and E. Verbitskiy, Homoclinic points, atoral polynomials, and periodic points of algebraic \(\mathbb{Z}^d\)-actions, Ergodic Theory Dynam. Systems 33 (2013), no. 4, 1060-1081.
- 27. D. A. Lind, K. Schmidt, and T. Ward, Mahler measure and entropy for commuting automorphisms of compact groups, Invent. Math. 101 (1990), no. 3, 593-629.
- 28. D. A. Lind and T. Ward, Automorphisms of solenoids and \(p\)-adic entropy, Ergodic Theory Dynam. Systems 8 (1988), no. 3, 411-419.
- 29. D. W. Masser, Mixing and linear equations over groups in positive characteristic, Israel J. Math. 142 (2004), 189-204.
- 30. K. Schmidt, Mixing automorphisms of compact groups and a theorem by Kurt Mahler, Pacific J. Math. 137 (1989), no. 2, 371-385.
- 31. K. Schmidt, Automorphisms of compact abelian groups and affine varieties, Proc. London Math. Soc. (3) 61 (1990), no. 3, 480-496.
- 32. K. Schmidt, Dynamical systems of algebraic origin, Birkhäuser, Basel, 1995.
- 33. K. Schmidt and T. Ward, Mixing automorphisms of compact groups and a theorem of Schlickewei, Invent. Math. 111 (1993), no. 1, 69-76.
- 34. Ja. Sinaĭ, Flows with finite entropy, Dokl. Akad. Nauk SSSR 125 (1959), 1200-1202.
- 35. Ja. Sinaĭ, On the concept of entropy for a dynamic system, Dokl. Akad. Nauk SSSR 124 (1959), 768-771.
- 36. A. J. van der Poorten and H. P. Schlickewei, Additive relations in fields, J. Aust. Math. Soc. Ser. A 51 (1991), no. 1, 154-170.
- 37. P. Walters, An introduction to ergodic theory, Springer, New York, 1982.
- 38. S. A. Yuzvinskiĭ, Computing the entropy of a group of endomorphisms, Sib. Math. J. 8 (1967), 172-178.

Lind, Douglas

Department of Mathematics, University of Washington, Seattle, Washington, 98195, United States Schmidt, Klaus

Mathematics Institute, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria