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 ?