I am studying the basics on Ergodic Theory from Fundamentos da Teoria Ergodica by Oliveira & Viana (you can actually find the pdf on their website), particularly, the properties of topological pressure. I got a bit confused by some remarks on the join of partitions.
Let $f:M\to M$ continuous on a compact metric space $M$. If $\alpha$ es an open cover of $M$, then $f^{-j}(\alpha)$ is also an open cover for every $i\in\mathbb{N}$. Define $\alpha^n$ by
$$ \alpha^n:=\bigvee_{i=0}^{n-1}f^{-i}(\alpha)=\{P_0\cap...\cap P_n : P_i\in f^{-j}(\alpha)\}$$
At page 301, it s remarked that $(\alpha^k)^n=\alpha^{n+k}$. But then at page 302, it is remarked that $(\alpha^k)^n=\alpha^{kn}$. I got really confused, so I tried to prove by myself.
I could see that $f^{-1}(\bigvee_{i}P_i)=\bigvee f^{-1}(P_i)$, so by iterating you get that
\begin{align*} (\alpha^k)^n&=\left( \bigvee_{j=0}^{k-1}f^{-j}(\alpha)\right)^n \\ &=\bigvee_{l=0}^{n-1}f^{-l}\left(\bigvee_{j=0}^{k-1}f^{-j}(\alpha) \right)\\ &=\bigvee_{l=0}^{n-1}\bigvee_{j=0}^{k-1}f^{-(j+l)}(\alpha) \end{align*}
and I don't know how to get $(\alpha)^{kn}$ or $(\alpha)^{n+k}$ from that.
Edit
The context I'm working is proving the following properties of topological pressure: $P(f,\phi,\alpha^k)=P(f,\phi,\alpha)$ and $P(f^k,\phi_k)=k P(f,\phi)$ with the open cover definition of pressure. Is there any way to prove it without using those remarks? (Using the open cover definition of pressure, not the $(n,\epsilon)$ definition used by Walters).