# a topological mixing example

A continuous map $$f:X\to X$$ is called Topologically transitive or TT if for every pair of non-empty open sets $$U,V$$ in $$X$$ there exists $$n\in \Bbb{N}$$ such that $$f^n(U)\cap V \neq \emptyset$$.

A continuous map $$f:X\to X$$ is called weak mixing if $$f\times f$$ is TT.

and

A continuous map $$f:X\to X$$ is called Topological Mixing if for every pair of non-empty open sets $$U,V$$ in $$X$$ there exists $$n_0\in \Bbb{N}$$ such that $$f^n(U)\cap V \neq \emptyset\ \forall\ n\ge n_0$$.

We can prove that Topological Mixing $$\implies$$ Weak mixing $$\implies$$ TT.
I have a map which is topologically mixing, but I don't know how to show this ?
This is the map :
\begin{align*} &g : [0,1] \longrightarrow [0,1] \\ &g(x)=3((\dfrac{x-1}{3})- \vert x-\dfrac{1}{3} \vert + \vert x - \dfrac{2}{3}\vert ) \end{align*} Would you mind helping me ?

• $g$ doesn't take values in $[0,1]$. For instance $g(1)=-1$. – Yanko Aug 21 '19 at 16:50