Sequence entropy and mixing

The relationship between sequence entropy and mixing is examined. Let T be an automorphism of a Lebesgue space X, $\text{L}$₀ denote the set of all partitions of X possessing finite entropy, and $\text{S}$ denote the set of all increasing sequences of positive integers. It is shown that: (1) T is mixing /a2 sup_{A ? B}h_A(T, α) = H(α) for all B∈I and α∈Z₀. (2) T is weakly mixing /a2 sup_Ah_A(T, α) = H(α) for all α∈Z₀. (3) If T is partially mixing with constant $\text{c (1 ?}\text{1}\text{e}\text{< c < 1)}$, then sup_{A ? B}h_A(T, α) > cH(α) for all B∈I and nontrivial α∈Z₀. (4) If sup_{A ? B}h_A(T, α) > 0 for all B∈I and nontrivial α∈Z₀, then T is weakly mixing.