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For any measure preserving system $(X,mathcal{B},mu,T_1,ldots,T_d),$ where we assume no commutativity on the transformations $T_i,$ $1leq ileq d,$ we study the pointwise convergence of multiple ergodic averages with iterates of different growth coming from a large class of sublinear functions. This class properly contains important subclasses of Hardy field functions of order $0$ and of Fejer functions, i.e., tempered functions of order $0.$ We show that the convergence of the single average, via an invariant property, implies the convergence of the multiple one. We also provide examples of sublinear functions which are in general bad for convergence on arbitrary systems, but they are good for uniquely ergodic systems. The case where the fastest function is linear is addressed as well, and we provide, in all the cases, an explicit formula of the limit function.
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Exploiting the recent work of Tao and Ziegler on the concatenation theorem on factors, we find explicit characteristic factors for multiple averages along polynomials on systems with commuting transformations, and use them to study the criteria of joint ergodicity for sequences of the form $(T^{p_{1,j}(n)}_{1}cdotldotscdot T^{p_{d,j}(n)}_{d})_{ninmathbb{Z}},$ $1leq jleq k$, where $T_{1},dots,T_{d}$ are commuting measure preserving transformations on a probability measure space and $p_{i,j}$ are integer polynomials. To be more precise, we provide a sufficient condition for such sequences to be jointly ergodic. We also give a characterization for sequences of the form $(T^{p(n)}_{i})_{ninmathbb{Z}}, 1leq ileq d$ to be jointly ergodic, answering a question due to Bergelson.
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