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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2Seems to me you won't be able to do both... – Robert Israel Sep 24 '12 at 06:21
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(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. – Sep 24 '12 at 16:21
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Robert Israel can you tell me which corollary of the Hartman-Grobman theorem? – Oct 04 '13 at 13:38
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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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Ok, $0$ is a fixed point. But why is $0$ hyperbolic? Shouldn't the function be differentiable at the point? – Odylo Abdalla Costa Apr 07 '20 at 02:53
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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.
Mathowenw
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