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Mixing automorphisms of compact groups and a theorem of Schlickewei. (English) Zbl 0824.28012

The authors show that if \(d>1\) and \(\alpha\) is a mixing \(\mathbb Z^ d\)- action by automorphisms of a connected compact Abelian group \(X\), then \(\alpha\) is mixing of all orders. In particular this is true for commuting toral automorphisms. The result is known to be false if \(X\) is disconnected. The decidedly nontrivial proof begins by establishing an algebraic characterization of \(r\)-mixing in terms of certain prime ideals in the ring of Laurent polynomials \(\mathbb Z[u^{\pm 1}_ 1,\dots, u^{\pm 1}_ d]\). Then they apply a deep theorem of H. P. Schlickewei [Invent. Math. 102, No. 1, 95–107 (1990; Zbl 0711.11017)] which gives a bound on the number of solutions in \(S\)-units of an algebraic number field of the equation \(a_ 1 v_ 1+\cdots+ a_ n v_ n= 1\) for which no proper subsum vanishes.

MSC:

28D15 General groups of measure-preserving transformations
22D40 Ergodic theory on groups
37A25 Ergodicity, mixing, rates of mixing
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)

Citations:

Zbl 0711.11017
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References:

[1] [AM] Atiyah, M., MacDonald, I.G.: Introduction to Commutative Algebra. Reading, MA: Addison-Wesley 1969 · Zbl 0175.03601
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[3] [KS2] Kitchens, B., Schmidt, K.: Mixing Sets and Relative Entropies for Higher Dimensional Markov Shifts. (Preprint 1991)
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[5] [Le] Ledrappier, F.: Un champ markovien peut être d’entropie nulle et mélangeant. C.R. Acad. Sci. Paris, Ser.A 287, 561-562 (1978) · Zbl 0387.60084
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[8] [MT] Miles, G., Thomas, R.K.: The breakdown of automorphisms of compact topological groups. In: Studies in Probability and Ergodic Theory. (Adv. Math. Suppl. Stud., vol. 2, pp. 207-218) New York London: Academic Press 1987
[9] [S] Schlickewei, H.P.:S-unit equations over number fields. Invent. Math.102, 95-107 (1990) · Zbl 0711.11017
[10] [Sc1] Schmidt, K.: Automorphisms of compact abelian groups and affine varieties. Proc. Lond. Math. Soc.61, 480-496 (1990) · Zbl 0789.28013
[11] [Sc2] Schmidt, K.: Mixing automorphisms of compact groups and a theorem by Kurt Mahler. Pac. J. Math.137, 371-384 (1989) · Zbl 0678.22002
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