Asymmetries in Asymptotic 3-fold Properties of Ergodic Actions

Abstract

We present: 1) a mixing $Z ^ 2$-action with the following asymmetry of multiple mixing property: for some commuting measure-preserving transformations $S$, $T$ and a sequence $n_j$ $$ \lim_{j\to \infty}\mu(A\bigcap S^{-n_j}A\bigcap T^{-n_j}A)=\mu(A)^3$$ for all measurable sets $A$, but there is $A_0$, $\mu(A_0)=\frac 1 2$, such that $$ \lim_{j\to \infty}\mu(A_0\bigcap S^{n_j}A_0\bigcap T^{n_j}A_0)=0;$$ 2) $Z $-actions with the asymmetry of the partial multiple mixing and the partial multiple rigidity: $$ \lim_{j\to \infty}\mu(A\bigcap T^{k_j}A\bigcap T^{m_j}A)= \frac23 \mu(A)^3+\frac13\mu(A),$$ $$ \lim_{j\to \infty}\mu(A\bigcap T^{-k_j}A\bigcap T^{-m_j}A)= \mu(A)^2;$$ 3) infinite transformations $T$ such that for all $A$, $\mu(A)<\infty$, $$\lim_{j\to \infty}\mu(A\bigcap T^{k_j}A\bigcap T^{m_j}A)= \frac13\mu(A)$$ and $$\lim_{j\to \infty}\mu(A\bigcap T^{-k_j}A\bigcap T^{-m_j}A)=0.$$