Let （X, f ） be a topological dynamical system, where X is a nonempty compact and metrizable space with the metric d and f ：X →X is a continuous map. For any integer n≥2, denote the product space by X（n）=X · · · × X｜ {z }n times . We say a system （X, f ） is generally distributionally n-chaotic if there exists a residual set D ？X（n） such that for any point x=（x1, · · · , xn）∈D, lim inf k→∞#（{i：0≤i≤k-1, min{d（fi（xj）, fi（xl））：1≤j 6=l≤n}〈δ0}） k=0 for some real numberδ0〉0 and lim sup k→∞#（{i：0≤i≤k-1, max{d（fi（xj）, fi（xl））：1≤j 6=l≤n}〈δ}） k=1 for any real number δ 〉 0, where #（·） means the cardinality of a set. In this paper, we show that for each integer n ≥ 2, there exists a system （X,σ） which satisfies the following conditions： （1） （X,σ） is transitive;（2） （X,σ） is generally distributionally n-chaotic, but has no distributionally （n＋1）-tuples;（3） the topological entropy of （X,σ） is zero and it has an IT-tuple.
In this paper,the relationship between the extended family and several mixing properties in measuretheoretical dynamical systems is investigated.The extended family eF related to a given family F can be regarded as the collection of all sets obtained as"piecewise shifted"members of F.For a measure preserving transformation T on a Lebesgue space(X,B,μ),the sets of"accurate intersections of order k"defined below are studied,Nε(A0,A1,...,Ak)=n∈Z+:μk i=0T inAiμ(A0)μ(A1)μ(Ak)<ε,for k∈N,A0,A1,...,Ak∈B and ε>0.It is shown that if T is weakly mixing(mildly mixing)then for any k∈N,all the sets Nε(A0,A1,...,Ak)have Banach density 1(are in(eFip),i.e.,the dual of the extended family related to IP-sets).
For each real number λ∈ [0, 1], λ-power distributional chaos has been in- troduced and studied via Furstenberg families recently. The chaoticity gets stronger and stronger as A varies from 1 to 0, where 1-power distributional chaos is exactly the usual distributional chaos. As a generalization of distributional n-chaos,λ-power distributional n-chaos is defined similarly. Lots of classic results on distributional chaos can be improved to be the versions of λ-power distributional n-chaos accordingly. A practical method for distinguishing 0-power distributional n-chaos is given. A transitive system is constructed to be 0-power distributionally n-chaotic but without any distributionally （n ＋ 1）-scrambled tuples. For each λ∈ [0, 1], ）,-power distributional n-chaos can still appear in minimal systems with zero topological entropy.
