Discrepancy of behavior of perturbed sequences in 𝐿^{𝑝} spaces

Abstract

Given p ∈ [ 1 , ∞ ) p \in [1,\infty ) , examples of sequences { n k } k ⊂ N {\{ {n_k}\} _{k \subset \mathbb {N}}} such that for any ergodic dynamical system ( X , β , m , T ) (X,\beta ,m,T) the averages \[ A N f ( x ) = 1 N ∑ k = 1 N f ( T n k x ) {A_N}f(x) = \frac {1} {N}\sum \limits _{k = 1}^N {f({T^{{n_k}}}x)} \] converge almost everywhere in all L q ( X ) , q > p {L^q}(X),\,q > p , but fail to have a finite limit for some function in L p ( X ) {L^p}(X) are shown. Also, sequences such that for all ergodic dynamical systems the averages A N f ( x ) {A_N}f(x) do not converge for some function f ∈ L p ( X ) f \in {L^p}(X) for all 1 ⩽ p > ∞ 1 \leqslant p > \infty but do converge for all functions in L ∞ ( X ) {L^\infty }(X) are shown.