On distributional n-chaos

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 We say a system (X,f) is generally distributionally n  -chaotic if there exists a residual set D ⊂ X(n)D ⊂ X(n) such that for any point x = (x1,…, xn) ∈ Dx = (x1,…, xn) ∈ D, lim infk→∞#({i : 0 ≤ i ≤ k −1, min{d(fi(xj), fi(xl)): 1 ≤ j ≠ l ≤ n}< δ0})k=0for some real number δ0 > 0 and lim supk→∞#({i : 0 ≤ i ≤ k −1, max{d(fi(xj), fi(xl)): 1 ≤ j ≠ l ≤ n}< δ0})k=1for 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.