A dynamical system is called a null system, if the topological sequence entropy along any strictly increasing sequence of non-negative integers is 0. Let 0≦p≦q≦1. A dynamical system is Dqp chaotic, if there is an u...A dynamical system is called a null system, if the topological sequence entropy along any strictly increasing sequence of non-negative integers is 0. Let 0≦p≦q≦1. A dynamical system is Dqp chaotic, if there is an uncountable subset in which any two different points have trajectory approaching time set with lower density p and upper density q. In this paper, we show that there is a null system which is also D3/41/4 chaotic.展开更多
Two infinite sequences A and B of non-negative integers are called infinite additive complements if their sum contains all sufficiently large integers. For a sequence T of non-negative integers, let T（x） be the numb...Two infinite sequences A and B of non-negative integers are called infinite additive complements if their sum contains all sufficiently large integers. For a sequence T of non-negative integers, let T（x） be the number of terms of T not exceeding x. In 1994, Sarkozy and Szemer′edi confirmed a conjecture of Danzer by proving that, for infinite additive complements A and B, if lim sup A（x）B（x）/x 1, then A（x）B（x）-x → ＋∞ as x → ＋∞. In this paper, it is proved that, if A and B are infinite additive complements with lim sup A（x）B（x）/x〈（√4 ＋ 2）/7 = 1.093 …, then（A（x）B（x）-x）/min{A（x）, B（x）} → ＋∞ as x → ＋∞.展开更多
基金supported by National Natural Science Foundation of China (Grant No.11071084)Natural Science Foundation of Guangdong Province (Grant No. 10451063101006332)
文摘A dynamical system is called a null system, if the topological sequence entropy along any strictly increasing sequence of non-negative integers is 0. Let 0≦p≦q≦1. A dynamical system is Dqp chaotic, if there is an uncountable subset in which any two different points have trajectory approaching time set with lower density p and upper density q. In this paper, we show that there is a null system which is also D3/41/4 chaotic.
基金supported by National Natural Science Foundation of China (Grant Nos. 11671211 and 11371195)the China Scholarship Council (Grant No. 201608320048)the Priority Academic Program Development of Jiangsu Higher Education Institutions
文摘Two infinite sequences A and B of non-negative integers are called infinite additive complements if their sum contains all sufficiently large integers. For a sequence T of non-negative integers, let T（x） be the number of terms of T not exceeding x. In 1994, Sarkozy and Szemer′edi confirmed a conjecture of Danzer by proving that, for infinite additive complements A and B, if lim sup A（x）B（x）/x 1, then A（x）B（x）-x → ＋∞ as x → ＋∞. In this paper, it is proved that, if A and B are infinite additive complements with lim sup A（x）B（x）/x〈（√4 ＋ 2）/7 = 1.093 …, then（A（x）B（x）-x）/min{A（x）, B（x）} → ＋∞ as x → ＋∞.
