离散动力系统上的一致收敛与传递集

Uniform Convergence and Transitive Sets on Discrete Dynamical Systems

摘要: 本文主要讨论离散动力系统上一致收敛与传递集的关系。设$\left(X,d\right)$ 是一个紧致度量空间，$f_n:X\to X$ 是$X$ 上的连续自映射序列，并一致收敛于一个连续自映射$f$ 。结果表明，当$\underset{m\to \infty }{lim}{d}_{\infty }\left(f_n^m,f^m\right)=0$ 时，若$X$ 的非空闭子集$A$ 是$\left(X,f_n\right)$ 的传递集，则$A$ 是$\left(X,f\right)$ 的传递集；若$X$ 的非空闭子集$A$ 是$\left(X,f_n\right)$ 的弱混合集，则$A$ 是$\left(X,f\right)$ 的弱混合集。

Abstract: In this paper, we mainly study the uniform convergence and transitive sets of discrete dynamical systems. Let X be a compact metric space with metric d, and $f_n:X\to X$ be a continuous self-map sequence on X such that $\left(f_n\right)$ converges uniformly to a continuous self-map f. It is shown that under the condition $\underset{m\to \infty }{lim}{d}_{\infty }\left(f_n^m,f^m\right)=0$ , let A be a closed subset of X, if A is a transitive set of $\left(X,f_n\right)$ , then A is a transitive set of $\left(X,f\right)$ ; if A is a weakly mixing set of $\left(X,f_n\right)$ , then A is a weakly mixing set of $\left(X,f\right)$ .