Rates of Divergence of non-Conventional Ergodic Averages

Details Rates of Divergence of non-Conventional Ergodic Averages Rates of Divergence of non-Conventional Ergodic Averages

2004-08-31

arxiv.org/abs/math/0409001v1

Mathematics

Dynamical Systems

25 pages, 1 figure

Scientific paper

We study the rate of growth of ergodic sums along a sequence (a_n) of times: S_N f(x)=f(T^{a_1}x) + ... + f(T^{a_N}x). We characterize the maximal rate of growth of these ergodic sums and identify a number of sequences such as (2^n) that achieve this rate of growth. We also return to Khintchine's strong uniform distribution Conjecture which stated that the averages (1/N)(f(x)+f(2x mod 1)+...+f(Nx mod 1)) converge pointwise almost everywhere to \int f for an integrable function on [0,1). We give an elementary counterexample to this conjecture, showing that divergence occurs at the maximal rate. In addition, we consider versions of Khintchine's conjecture in which the sequence of positive integers is replaced by the increasing enumeration of a subsemigroup of the positive integers. We give necessary and sufficient conditions on the semigroup for convergence in the modified conjecture.

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