Local stability of ergodic averages
نویسندگان
چکیده
منابع مشابه
Local Stability of Ergodic Averages
We consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages. It is not difficult to construct an example of a computable Lebesgue-measure preserving transformation of [0, 1] and a characteristic function f = χA such that the ergodic averages Anf do not converge to a computable element of L ([0, 1]). In particular, there is no computable bo...
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The case l = 1 is the mean ergodic theorem, and the result can be viewed as a generalization of that theorem. The l = 2 case was proven by Conze and Lesigne [Conze and Lesigne, 1984], and various special cases for higher l have been shown by Zhang [Zhang, 1996], Frantzikinakis and Kra [Frantzikinakis and Kra, 2005], Lesigne [Lesigne, 1993], and Host and Kra [Host and Kra, 2005]. Tao’s argument ...
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i.e., there exists a constant C such that SN(θ, u) ≤ CN . We define δ(θ, u) to be the infimum of the δ satisfying H1 for θ and u. About H1, in the case where θ takes its values in U (the set of complex numbers of modulus 1), it is clear that for all sequences θ and u, δ(θ, u) is smaller than or equal to 1 and it is well-known (see [Ka] for example) that it is greater than or equal to 1/2. Few e...
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We study the rate of growth of ergodic sums along a sequence (an) of times: SNf(x) = ∑ n≤N f(T nx). We characterize the maximal rate of growth and identify a number of sequences such as an = 2, along which the maximal rate of growth is achieved. We also return to Khintchine’s Strong Uniform Distribution Conjecture that the averages (1/N) ∑ n≤N f(nx mod 1) converge pointwise to ∫ f for integrabl...
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We show that for every ergodic system (X, μ,T1, . . . ,Td) with commuting transformations, the average 1 Nd+1 ∑ 0≤n1,...,nd≤N−1 ∑ 0≤n≤N−1 f1(T n 1 d ∏ j=1 T n j j x) f2(T n 2 d ∏ j=1 T n j j x) · · · fd(T n d d ∏ j=1 T n j j x). converges for μ-a.e. x ∈ X as N → ∞. If X is distal, we prove that the average 1 N N ∑ i=0 f1(T n 1 x) f2(T n 2 x) · · · fd(T n d x) converges for μ-a.e. x ∈ X as N → ∞...
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 2009
ISSN: 0002-9947
DOI: 10.1090/s0002-9947-09-04814-4