topologically transitive

A continuous surjection $f$ on a topological space $X$ to itselfis topologically transitive if for everypair of open sets $U$ and $V$ in $X$ there is an integer $n>0$ such that $f^{n}(U)\\cap V\eq\\emptyset$ , where $f^{n}$ denotes the $n$ -th iterate of $f$ .

If for every pair of open sets $U$ and $V$ there is an integer $N$ such that $f^{n}(U)\\cap V\eq\\emptyset$ for each $n>N$ , we say that $f$ is topologically mixing.

If $X$ is a compact metric space, then $f$ is topologically transitive if and only if there exists a point $x\\in X$ with a dense orbit, i.e. such that $\\mathcal{O}(x,f)=\\{f^{n}(x):n\\in\\mathbb{N}\\}$ is dense in $X$ .

单词	TopologicallyTransitive
释义	topologically transitive A continuous surjection $f$ on a topological space $X$ to itselfis topologically transitive if for everypair of open sets $U$ and $V$ in $X$ there is an integer $n>0$ such that $f^{n}(U)\\cap V\eq\\emptyset$ , where $f^{n}$ denotes the $n$ -th iterate of $f$ . If for every pair of open sets $U$ and $V$ there is an integer $N$ such that $f^{n}(U)\\cap V\eq\\emptyset$ for each $n>N$ , we say that $f$ is topologically mixing. If $X$ is a compact metric space, then $f$ is topologically transitive if and only if there exists a point $x\\in X$ with a dense orbit, i.e. such that $\\mathcal{O}(x,f)=\\{f^{n}(x):n\\in\\mathbb{N}\\}$ is dense in $X$ .
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