Can anyone help me on this exercise from Lan Wen's book? Here is what I did so far:
I've already proved that under that conditions, $f$ and $g_0$ are mixing. Then, we noticed that $F^n(x,y)=(f^n(x),g(f^{n-1}(x),g(f^{n-2}(x),...,g(x,y)))= (f^n(x),F^{n-1}(x,y))$.
To show that $F$ is mixing, consider the product topology and take $U_1\times V_1$ and $U_2\times V_2$ open subsets of $\mathbb{T}^2\times \mathbb{T}^2$.
$f$ is mixing, then for a sufficient large number $n>0$, $F^n(U_1\times V_1)\cap (U_2\times \mathbb{T}^2)\neq \emptyset$ for each of these $n$'s. Now, I have to use the fact that $g_0$ is mixing to show that $F$ is mixing. By $F^n(x,y)$ we see that $F\mid_{\{0\}\times\mathbb{T}^2}$ is mixing. But don't know how to use that to conclude. I guess the fact that $F^n(U_1\times V_1)\cap (U_2\times \mathbb{T}^2)\neq \emptyset$ is really important here because the intersection with $U_2$ will always happens so I don't need to worry about the first coordinate that much, once I have sufficient large $n$.
