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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 ?

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    $\begingroup$ $g$ doesn't take values in $[0,1]$. For instance $g(1)=-1$. $\endgroup$ – Yanko Aug 21 '19 at 16:50
  • $\begingroup$ @reza do you still care about this question? $\endgroup$ – mathworker21 Sep 6 '19 at 19:34
  • $\begingroup$ Yes I do....... $\endgroup$ – Reza Sep 7 '19 at 7:17

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