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Multiple return times theorems for weakly mixing systems. (English) Zbl 0985.37004
The author proves the pointwise convergence of the expression ${1\over N}\sum^N_{n=0} a_n g(R^n z)$ where $$(Z,K,\nu,R)$$ is an ergodic dynamical system on a probability measure space $$(Z,K,\nu)$$, the sequence of scalars $$a_n$$ has the form $a_n= a_n(x, y_1,y_2,\dots, y_J)= \Biggl(\prod^H_{i= 1} f_i(T^{b,n}x)\Biggr) \Biggl(\prod^J_{j=1} g_j(S^n_j y_j)\Biggr),$ $$(b_1,b_2,\dots, b_H)\in Z^H$$, $$J$$ is a positive integer, the functions $$f_i$$ and $$g_j$$ are bounded and $$(X,F,\mu, T)$$ and $$(Y,G_j, m_j, S_j)$$ are weakly mixing systems.

##### MSC:
 37A25 Ergodicity, mixing, rates of mixing 93E25 Computational methods in stochastic control (MSC2010)
##### Keywords:
ergodic dynamical system; weakly mixing systems
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