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A metric study involving independent sequences. (English) Zbl 0645.10044

This paper is devoted to the statistical study of sequences, taking values in a compact metric space \(X\), which have both properties of independence (i.e. statistical independence, Furstenberg’s concept of disjointness in ergodic theory and the (stronger) notion of spectral disjointness by Kamae) and well defined distribution. It may be considered as a continuation of the authors’ paper [Compos. Math. 51, 215–236 (1984; Zbl 0537.10030)], but from a different point of view involving metric concepts and dynamical systems as flows determined by sequences.
If \(u=(x_n)\in X^{\mathbb N}=Y\), consider \(S((x_0,x_1,x_2,\ldots))=(x_1,x_2,\ldots)\), \(u^*(n)=S^nu\) and \(K_u\) denotes the orbit closure of \(u^*\) with respect to \(S\), then \((K_u,S)\) (resp. \((K_u,S,\mu)\) is the flow determined by the sequence \(u\) (resp. the measured flow, if \(\mu\) is an invariant measure). In the first part general metric theorems are obtained expressing the statistical independence in terms of independence of associated spectral measures. A useful concept of energy spectrum of a sequence is introduced.
Then, following an idea of H. Furstenberg [Math. Syst. Theory 1, 1– 49 (1967; Zbl 0146.28502)] a very useful lemma on relative unique ergodicity of group extensions is proved having several interesting consequences (e.g. a new proof of the result of R. K. Thomas [Trans. Am. Math. Soc. 160, 103–117 (1971; Zbl 0222.28008)] that weakly mixing group extensions of a \(K\)-system are \(K\)-systems or an extension of a result of W. Parry [Am. J. Math. 91, 757–771 (1969; Zbl 0183.51503)],
In the final section \(G\) is a compact metric group, the splitting \(G\)-flow \(\Delta_G=(G^{\mathbb N},G)\) is given by the group action \((g,(g_0,g_1,\ldots))\to (gg_0,gg_1,\ldots)\). It is proved that \(u\) is a \(\Delta_G\) extension of \(\Delta u\) (i.e. this holds for the according flows, \(\Delta u=(x_n^{-1}x_{n+1})\), if \(u=(x_n))\) if and only if (i) \(u\) is uniformly distributed and (ii) \(u\) and \((\Delta u)^*\) are statistically independent.
Interesting applications on Weyl sequences \(P(n)=\sum^s_{k=0}a_kn^k\), \(s\geq 1\), \(a_s\) irrational, (which are extended to compact Abelian groups) are given (in the case of monothetic groups). It is proved f.i. that for any \(q\)-normal number and any real polynomial \(P(n)\) the sequences \((q^n\theta)\) and \((P(n))\) are completely statistically independent (generalizing the known case of \((2^n\theta)\) and \((n\alpha)\)). As applications several interesting results (independence and uniform distribution properties) on Bernoulli sequences and subsequences of \((n\alpha)\) are obtained.

MSC:

11K06 General theory of distribution modulo \(1\)
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
28D05 Measure-preserving transformations
22C05 Compact groups
22D40 Ergodic theory on groups
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
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References:

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