Asymmetries in Asymptotic 3-fold Properties of Ergodic Actions

We present:

a mixing $Z^{2}$ -action with the following asymmetry of multiple mixing property: for some commuting measure-preserving transformations $S$ , $T$ and a sequence $n_{j}$ $\lim_{j \to \infty} μ (A ⋂ S^{- n_{j}} A ⋂ T^{- n_{j}} A) = μ (A)^{3}$ for all measurable sets $A$ , but there is $A_{0}$ , $μ (A_{0}) = \frac{1}{2}$ , such that $\lim_{j \to \infty} μ (A_{0} ⋂ S^{n_{j}} A_{0} ⋂ T^{n_{j}} A_{0}) = 0;$
$Z$ -actions with the asymmetry of the partial multiple mixing and the partial multiple rigidity: $\lim_{j \to \infty} μ (A ⋂ T^{k_{j}} A ⋂ T^{m_{j}} A) = \frac{2}{3} μ (A)^{3} + \frac{1}{3} μ (A),$ $\lim_{j \to \infty} μ (A ⋂ T^{- k_{j}} A ⋂ T^{- m_{j}} A) = μ (A)^{2};$
infinite transformations $T$ such that for all $A$ , $μ (A) < \infty$ , $\lim_{j \to \infty} μ (A ⋂ T^{k_{j}} A ⋂ T^{m_{j}} A) = \frac{1}{3} μ (A)$ and $\lim_{j \to \infty} μ (A ⋂ T^{- k_{j}} A ⋂ T^{- m_{j}} A) = 0.$

We present:

a mixing $Z^{2}$ -action with the following asymmetry of multiple mixing property: for some commuting measure-preserving transformations $S$ , $T$ and a sequence $n_{j}$ $\lim_{j \to \infty} μ (A ⋂ S^{- n_{j}} A ⋂ T^{- n_{j}} A) = μ (A)^{3}$ for all measurable sets $A$ , but there is $A_{0}$ , $μ (A_{0}) = \frac{1}{2}$ , such that $\lim_{j \to \infty} μ (A_{0} ⋂ S^{n_{j}} A_{0} ⋂ T^{n_{j}} A_{0}) = 0;$
$Z$ -actions with the asymmetry of the partial multiple mixing and the partial multiple rigidity: $\lim_{j \to \infty} μ (A ⋂ T^{k_{j}} A ⋂ T^{m_{j}} A) = \frac{2}{3} μ (A)^{3} + \frac{1}{3} μ (A),$ $\lim_{j \to \infty} μ (A ⋂ T^{- k_{j}} A ⋂ T^{- m_{j}} A) = μ (A)^{2};$
infinite transformations $T$ such that for all $A$ , $μ (A) < \infty$ , $\lim_{j \to \infty} μ (A ⋂ T^{k_{j}} A ⋂ T^{m_{j}} A) = \frac{1}{3} μ (A)$ and $\lim_{j \to \infty} μ (A ⋂ T^{- k_{j}} A ⋂ T^{- m_{j}} A) = 0.$

文献摘要