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Fully generic sequences and a multiple-term return-times theorem. (English) Zbl 0910.28013
Let $(X,{\cal F},\mu, T)$ be a dynamical systems consisting of a measure-preserving transformation defined on a standard probability space. A bounded sequence $\{a_i\}$ of complex numbers are called universal weights (for the pointwise ergodic theorem) if for any dynamical system $(Y,{\cal G},\nu, S)$ and $g\in L^1(\nu)$, ${1\over N} \sum^{N-1}_{j= 0}a_ig(S^j y)$ converges. The following theorem was proved by {\it J. Bourgain}, {\it H. Furstenberg}, {\it Y. Katznelson} and {\it D. S. Ornstein} [Publ. Math., Inst. Haut. √Čtud. Sci. 69, 5-45 (1989; Zbl 0705.28008)] (and a joinings proof was given by {\it D. J. Rudolph} [Ergodic Theory Dyn. Syst. 14, No. 1, 197-203 (1994; Zbl 0799.28010)]): For any dynamical system $(X,{\cal F},\mu,T)$ and $f\in L^\infty(\mu)$ for $\mu$ a.e. $x$, the sequence of values $f(T^ix)$ are universal weights. The author proves a multi-term version of the above result first proposed by I. Assani, i.e., involving averages of the form $${1\over N} \sum^{N-1}_{i= 0} f_1(T^i_1 x_1)f_2(T^i_2 x_2)\cdots f_k(T^i_k x_k),$$ where all the $f_i$ are bounded and where for each $j<k$, the points $x_1,x_2,\dots, x_j$ guaranteeing convergence can be chosen universally, without knowledge of the transformations and functions to follow. The author is able to avoid the $L^2$ orthogonality arguments in the proof of Bourgain et al., which requires the splitting of the $L^2$ space into Kronecker functions and their orthocomplement, by using the notion of pointwise genericity. In particular, a disjointness result on joinings avoids the need to identify distinguished factor algebras for the higher term averages. A disadvantage of this approach is that delicate information about characteristic factors is not available (as in, for example, the work of {\it D. J. Rudolph} [Lond. Math. Soc. Lect. Note Ser. 205, 369-432 (1995; Zbl 0877.28012)], concerning the Conze-Lesigne algebra).

28D05Measure-preserving transformations
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