具有${\bm{\mathscr M}}_{\bm\alpha}$-跟踪性质的动力系统的弱混合

中文摘要 AI 辅助

本文利用弱散射的泛因子刻画，研究具有$\mathscr{M}_{\alpha}$-跟踪性质的两类动力系统的弱混合性。首先，对每个$\alpha\in[0,1)$，每个具有$\mathscr{M}_{\alpha}$-跟踪性质的$M$-系统都是弱混合的，从而否定了[P. Oprocha等，On partial shadowing of complete pseudo-orbits, J. Math. Anal. Appl., Question~1]中的问题。其次，对每个$\alpha\in(0,1)$，$\mathscr{M}_{\alpha}$-跟踪性质蕴含$E$-系统的弱混合性，特别是对测度中心为整个空间的动力系统。最后，构造了一个具有$\mathscr{M}_{0}$-跟踪性质但不是弱混合的$E$-系统，表明$E$-系统结果中参数范围$\alpha\in(0,1)$是最优的。

英文摘要

This paper studies weak mixing for two classes of dynamical systems with the \(\mathscr{M}_α\)-shadowing property using the generic factor characterization of weak scattering. First, for every \(α\in[0,1)\), every \(M\)-system with the \(\mathscr{M}_α\)-shadowing property is weakly mixing, thereby answering [P. Oprocha et. al., On partial shadowing of complete pseudo-orbits, J. Math. Anal. Appl., Question~1] negatively. Second, for every \(α\in(0,1)\), the \(\mathscr{M}_α\)-shadowing property implies weak mixing for \(E\)-systems and, in particular, for dynamical systems whose measure center is the whole space.Finally, an \(E\)-system with the \(\mathscr{M}_{0}\)-shadowing property which is not weakly mixing is constructed, showing that the parameter range \(α\in(0,1)\) in the \(E\)-system result is optimal.

发表机构

• School of Mathematics and Statistics, Guizhou University of Finance and Economics（贵州财经大学数学与统计学院）
• Department of Mathematics, Shandong University, Weihai（山东大学威海数学系）

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