if $f$ is weakly mixing then $f^n$ is ergodic?

Question

If $f$ is weakly mixing then $f^n$ is ergodic? I think this is false but I can't find a counter-example because I don't know transformations weakly mixing but not mixing. Can you prove or give a counterexample?

Answer 1

That $(f,\mu)$ is weakly mixing means that $$ \frac{1}{N} \sum_{k=0}^{N-1} |\mu (A \cap f^{-k} B) - \mu(A)\mu(B)| \to 0. $$ Similarly $(f^n,\mu)$ is weakly mixing if $$ \frac{1}{N} \sum_{k=0}^{N-1} |\mu (A \cap f^{-kn} B) - \mu(A)\mu(B)| \to 0. $$

If you compare $$ \frac{1}{N} \sum_{k=0}^{N-1} |\mu (A \cap f^{-kn} B) - \mu(A)\mu(B)| $$ and $$ \frac{1}{nN} \sum_{k=0}^{nN-1} |\mu (A \cap f^{-k} B) - \mu(A)\mu(B)| $$ you will see that $(f^n,\mu)$ is weakly mixing if $(f,\mu)$ is. Finally, observe that weakly mixing implies ergodicity.