Periodic point near Hyperbolic fixed point

Solution 1:

You can do it by contradiction.

Suppose this is not true. You will find a $M$ and a sequence of point $p_k \to p$ of periodic point converging to $p$ with period less than $M$.

Taking a sub-sequence, they can have the same period $m \leq M$ so they are fixed for $f^m$. Notice that $p$ is still a hyperbolic fixed point of $f^m$.

Now taking again a sub-sequence, they can approch $p$ in the same direction, that is $\frac{p-p_k}{\| p -p_k \|} \to v \in E$.

Computing the differential in the $v$ direction for $f^m$ with this sequence will give you a $0$ and this cannnot be since it is suppose hyperbolic.