一个无界混沌的例子
An example of unbounded chaos
摘要：
设$ϕ(x) = |1 - \frac 1x|$为所有$x > 0$。然后，我们扩展$ϕ(x)$在通常的方式成为一个连续的映射，从紧拓扑（但不是度量）空间$[0, \infty]$到自身也映射的集合无理点在$(0, \infty)$到自身。在本文中，我们证明了（1）在$[0, \infty]$上，$ϕ(x)$是拓扑混合的，具有稠密的无理周期点，并且具有拓扑熵$\log λ$，其中$λ$是多项式$x^3 - 2x -1$的唯一正零;（2）$ϕ(x)$在$(0, 3)$中具有有界不可数{\it不变量} 2-扰码无理点集;（3）对任意可数无限点集$X$（有理数或无理数），存在$(0, \infty)$中无理传递点的稠密无界不可数{\it不变量} $\infty$-乱集$Y$，使得对于任何$x \in X$和任何$y \in Y$，我们有$\limsup_{n \to \infty} |ϕ^n(x) - ϕ^n(y)| = \infty$和$\liminf_{n \to \infty} |ϕ^n(x) - ϕ^n(y)| = 0$。这证明了$ϕ(x)$混沌的真正本质。
英文摘要：
Let $ϕ(x) = |1 - \frac 1x|$ for all $x > 0$. Then we extend $ϕ(x)$ in the usual way to become a continuous map from the compact topological (but not metric) space $[0, \infty]$ onto itself which also maps the set of irrational points in $(0, \infty)$ onto itself. In this note, we show that (1) on $[0, \infty]$, $ϕ(x)$ is topologically mixing, has dense irrational periodic points, and has topological entropy $\log λ$, where $λ$ is the unique positive zero of the polynomial $x^3 - 2x -1$; (2) $ϕ(x)$ has bounded uncountable {\it invariant} 2-scrambled sets of irrational points in $(0, 3)$; (3) for any countably infinite set $X$ of points (rational or irrational) in $(0, \infty)$, there exists a dense unbounded uncountable {\it invariant} $\infty$-scrambled set $Y$ of irrational transitive points in $(0, \infty)$ such that, for any $x \in X$ and any $y \in Y$, we have $\limsup_{n \to \infty} |ϕ^n(x) - ϕ^n(y)| = \infty$ and $\liminf_{n \to \infty} |ϕ^n(x) - ϕ^n(y)| = 0$. This demonstrates the true nature of chaos for $ϕ(x)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
