1.Suppose f is a diffeomorphism.Prove that all hyperbolic periodic points are isolated.
2.Show via an example that hyperbolic periodic points need not be isolated.
1.Suppose f is a diffeomorphism.Prove that all hyperbolic periodic points are isolated.
2.Show via an example that hyperbolic periodic points need not be isolated.
As for the second question consider $ f(x)=\begin{cases}2x\sin(x^{-1}) &\quad\text{ if }\quad x\neq 0\\0&\quad\text{ if }\quad x= 0\end{cases} $
then $0$ is the limit of a sequence of hyperbolic fixed points.
You can use the Hartman-Grobman theorem's corollary to solve this problem. More specifically, you can use its corollary about the local stable and unstable manifolds to solve this problem.