Theorem 2.1. Let $X$ be a compact metric space and $f:X\to X$ be a continuous map. Then the following are equivalent:

(1) $f$ is a topological mixing map;

(2) $f$ is a topological transitive map and for any non-empty open sets $U,V\subset X$, there exists $n_0\in\mathbb{N}$ such that $f^n(U)\cap V\neq\emptyset$ for all $n\geq n_0$.

(3) $f$ is a topological transitive map and for any non-empty open sets $U,V\linebreak U,V\subset X$, there exists $n_0\in\mathbb{N}$ such that $f^n(U)\cap V\neq\emptyset$ for all $n\geq n_0$ and $f^n(V)\cap U\neq\emptyset$ for all $n\geq n_0$.

Proof. (1)$\Rightarrow$(2) is trivial. (2)$\Rightarrow$(3) is obvious. (3)$\Rightarrow$(1) is clear. ∎

Theorem 2.2. Let $X$ be a compact metric space and $f:X\to X$ be a continuous map. Then the following are equivalent:

(1) $f$ is a topological mixing map;

(2 map;

(2 map;

*(2.