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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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    Robert Israel can you tell me which corollary of the Hartman-Grobman theorem?2013-10-04

2 Answers 2

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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?)2014-06-17