Let $f:M\to M$ a continuous in a compact metric space. For each $\phi:M\to\mathbb{R}$ and $n\in\mathbb{N}$, define $\phi_n:M\to\mathbb{R}$ by $$ \phi_n =\sum_{i=0}^{n-1}\phi\circ f^i $$ For a non-empty $C\subseteq M$, denote $$ \phi_n(C)=\sup\{\phi_n (x) : \ x\in C \} $$ For each open cover $\alpha$ of $M$, denote $\alpha^n=\displaystyle\bigvee_{i=0}^{-n+1}f^{-i}(\alpha)$. Finally, define $$ P_n(f,\phi,\alpha)=\inf\{ \sum_{U\in\gamma} e^{\phi_n(U)}:\gamma \mbox{ finite subcover of } \alpha^n\} $$ Prove that the sequence $\{\log P_n(f,\phi,\alpha)\}_{n\in\mathbb{N}}$ is subadditive.
My problem is that I have no idea how to compare $e^{\phi_{n+m}(U)}$ with $e^{\phi_n(U)}$ and $e^{\phi_m(U)}$. Any help is appreciated. Thanks in advance!
Take a finite subcover $\gamma_1\subseteq\alpha^n$ that $\varepsilon$-realizes the infimum in $P_n(f,\phi,\alpha)$ and a finite subcover $\gamma_2\subseteq\alpha^m$ that $\varepsilon$-realizes the infimum in $P_m(f,\phi,\alpha)$, and try the finite subcover $\gamma_1\land f^{-m}\gamma_2$ of $\alpha^{m+n}$. Use the fact that $$\sum_{U\in\gamma_2} e^{\phi_m(U)} = \sum_{U'\in f^{-m}\gamma_2}e^{\phi_m\circ f^m(U')}$$ and that $\phi_k(U)$ is non-decreasing in $U$.