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Abstract

If the ergodic transformations S, T generate a free $ℤ^2$ action on a finite non-atomic measure space (X,S,µ) then for any $c_1,c_2 ∈ ℝ$ there exists a measurable function f on X for which $({N+1})^{-1} ∑_{j=0}^Nf(S^jx) → c_1$ and $(N+1)^{-1} ∑_{j=0}^Nf(T^jx) → c_2 µ$-almost everywhere as N → ∞. In the special case when S, T are rationally independent rotations of the circle this result answers a question of M. Laczkovich.