Mixing via the extended family

Abstract

In this paper, the relationship between the extended family and several mixing properties in measuretheoretical dynamical systems is investigated. The extended family $e\mathcal{F}$ related to a given family $\mathcal{F}$ can be regarded as the collection of all sets obtained as “piecewise shifted” members of $\mathcal{F}$. For a measure preserving transformation T on a Lebesgue space $\left( {X,\mathcal{B},\mu } \right)$, the sets of “accurate intersections of order k” defined below are studied,

$$N_\varepsilon \left( {A_0 ,A_1 , \ldots ,A_k } \right) = \left\{ {n \in \mathbb{Z}_ + :\left| {\mu \left( {\bigcap\limits_{i = 0}^k {T^{ - in} A_i } } \right) - \mu \left( {A_0 } \right)\mu \left( {A_1 } \right) \cdots \mu \left( {A_k } \right)} \right| < \varepsilon } \right\},$$

for k ∈ ℕ, $A_0 ,A_1 , \ldots ,A_k \in \mathcal{B}$ and ɛ > 0. It is shown that if T is weakly mixing (mildly mixing) then for any k ∈ ℕ, all the sets N _ε (A ₀, A ₁, …, A _k ) have Banach density 1 (are in $\left( {e\mathcal{F}_{ip} } \right)^*$, i.e., the dual of the extended family related to IP-sets).