A counterexample on multiple convergence without commutativity

It is shown that there exists a probability space ( X , X , μ ) (X,{ X},μ ) , two ergodic measure preserving transformations T , S T,S acting on ( X , X , μ ) (X,{ X},μ ) with h μ ( X , T ) = h μ ( X , S ) = 0 h_μ (X,T)=h_μ (X,S)=0 , and f , g ∈ L ∞ ( X , μ ) f, g ∈ L^∞ (X,μ ) such that the limit lim N → ∞ 1 N ∑ n = 0 N − 1 f ( T n x ) g ( S n x ) {equation*} lim N→ ∞1/N∑ ₙ₌₀N-1 f(Tⁿx)g(Sⁿx) {equation*} does not exist in L 2 ( X , μ ) L^2(X,μ ) .