Let $f: M \to M$ to be a diffeomorphism where $M$ is a finite dimensional compact manifold. We say that a fixed point $p$ (i.e., $f(p)=p$) is elementary if $1 \notin sp(Df_{p})$. Suppose that every fixed point of $M$ is a elementary fixed point.
Claim: There exists only a finite number of elementary fixed points.
I tried to show that the elementary points is discret set, but I am stuck in this.
The statement is from Robinson's Dynamical systems.
Thanks.
Let us deal with the problem locally. Let $p\in M$ be an elementary fixed point. Using charts, you can assume that $p=0$ and that $f(0)=0$. If $1\notin\operatorname{Sp}(Df_0)$, then $D(f-\operatorname{Id})_0$ is invertible. So, by the inverse function theorem, $f-\operatorname{Id}$ is injective in some neighborhood $U$ of $0$. But then, since $(f-\operatorname{Id})(0)=0$, if $q\in U\setminus\{0\}$, then $(f-\operatorname{Id})(q)\neq q$. In other words, $f(q)\neq q$. So, near $0$ there aro no other fixed points. Reverting to the original situation, you deduce that near $p$ there are no other fixed points.
Therefore, the elementary fixed points form a discrete set.