Let $f:X\to X$ be a homeomorphism on compact metric space $(X, d)$ with two following property:
1) Every minimal set of $X$ is a fixed point i.e. if $K\subseteq X$ is a closed $f$-invariant set with $\overline{O_f(a)}=K$ for all $a\in K$, then $K=\{p\}$.
2) $f$ has finitely fixed point.
It is known that every closed $f$-invariant set $A$ contains a minimal set, hence I think that (1) implies that $\omega_f(x)=\{p\}$. Take $\omega_f(B)=\bigcup_{b\in B}\omega_f(b)$, I think that by (2), we have $\omega_f(B)$ is a finite set.
What can say about $f:X\to X$. Can we say that $X$ is countable or $f$ has finite orbit? Would you please your idea about them.