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.
We have a map that is TT but not weak mixing as
$f(x)=\begin{cases} 2x+1 & 0\le\ x\ \le\ \frac{1}{2} \\ -2x+3 & \frac{1}{2}\le\ x\ \le\ 1 \\ -x+2 & 1\le x\le 2 \end{cases}\\ $.
And irrational rotations of $S^1$ provides a map which is TT but not Topological Mixing.
I am looking for an example of a map which is a weak mixing but not Topological Mixing.
Any help here is appreciated. Thanks in advance.