diffeomorphism and hyperbolic periodic points

Question

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) 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.

Norbert · Answer 1 · 2012-10-15 15:53:54Z

up vote 1 down vote

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.

edited Oct 15 '12 at 15:53

Norbert
20.5k42270

answered Oct 14 '12 at 18:57

mahmud
111

1 Answer