关键词: transitive systems, distributional chaos

Abstract:

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. 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, · · · , x_n) ∈ D,
liminf_k→∞#({i : 0 ≤ i ≤ k − 1,min{d(f ⁱ(x_j), f ⁱ(x_l)) : 1 ≤ j ≠ l ≤ n} < δ₀})/k= 0
for some real number δ₀ > 0 and

limsupf_k→∞#({i : 0 ≤ i ≤ k − 1,max{d(f ⁱ(x_j), f ⁱ(x_l)) : 1 ≤ j ≠ l ≤ n} < δ₀})/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.

Key words: transitive systems, distributional chaos