# diffeomorphism and hyperbolic periodic points

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.

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Seems to me you won't be able to do both... –  Robert Israel Sep 24 '12 at 6:21
(1) is a corollary of the Hartman-Grobman theorem. (2) should probably be "non-hyperbolic periodic points need not be isolated". Then, the example is trivial. –  user8126 Sep 24 '12 at 16:21
Robert Israel can you tell me which corollary of the Hartman-Grobman theorem? –  user98737 Oct 4 '13 at 13:38

## 2 Answers

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.

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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.

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Where can one find those corollaries? (Where you meet them?) –  Hoseyn Heydari Jun 17 at 6:43