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 ...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.展开更多
基金supported by the NNSF of China(11071084,11026095 and 11201157)FDYT of Guangdong Province(2012LYM 0133)
文摘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.
