Ergodic averages of commuting transformations with distinct degree polynomial iterates

Abstract

Abstract. We prove mean convergence, as N → ∞, for the multiple ergodic averages

Integer part polynomial correlation sequences

“…Also, in some special cases of Theorem 1.4, via the theory of characteristic factors (equivalently, via the seminorms ||| · ||| k ) using Theorem 1.2 and Proposition 5.1 from [6], we prove convergence to 0 for the previous averages. More specifically, we prove the following (for the definitions, see Section 2): Theorem 1.5.…”

Integer part independent polynomial averages and applications along primes

“…The previous remark shows that (5) typically fails even for families of independent, integer polynomials. Hence, Theorem 2.2 is another indication that one has to work with real polynomials in order to have nice lower bounds as in (5) for general systems.…”

“…It was also shown in [17] that for every k ∈ N the factor Z k has an algebraic structure, in fact we can assume that it is a k-step nilsystem. This is the content of the following Structure theorem, which we recall in the ergodic case: 5 Equivalently, if E(fi|Y) = 0 for some 1 ≤ i ≤ ℓ, then lim…”

“…The convergence of multiple ergodic averages with several commuting transformations has been studied as well. Under weaker assumptions to the weakly mixing one on the system, the convergence to the "right"-expected limit, for the case of iterates of integer polynomials with different degrees and of integer part of real polynomials with different degrees, is treated in [5] and [18] respectively, where by right limit we mean that under the ergodic assumption, the limit is equal to the product of integrals of f i .…”

Poincaré recurrence and number theory: thirty years later

“…The case when all the transformations T i are equal with linear exponents is fully understood, with partial results (for example in [21] and [15]) and the complete convergence (in [32]). These results have been generalized and viewed in other ways, with further studies of the linear case [49], polynomial iterates ( [27], [33] and [41]), commuting transformations ( [16] and [46]), restrictions on the iterates for commuting transformations (see for example [36], [14], [1], and [2]), nilpotent group actions [9], and the corresponding average for flows (see [43], [10] and [3]). For a single transformation, we have a complete understanding in [32] of the structures controlling convergence (with a topological analog in [35]).…”