Let $f:M\rightarrow M$ be a $C^{1}$-class diffeomorphism . Let $x\in M$ be a fixed point.
I've been looking for a while on Internet for a proof of the following fact, but i couldn't find :
$\lbrace x\rbrace$ is a hyperbolic set for $f$ if and only if $x$ is a hyperbolic fixed point.
The definition of hyperbolic fixed point I'm using is the following : $x$ is a fixed point of $f$ such that $d_{x}f:T_{x}M\rightarrow T_{x}M$ has no eigenvalues in the unit circle $S^{1}\subset\mathbb{C}$.
Can somebody help me ? (sketching the proof or even giving me some reference)
Thank you 🙂
Best Answer
This is simply a matter of definition: a hyperbolic fixed point is defined to be a point $x$ such that $\{x\}$ is a hyperbolic set. See for example the wikipedia entry on hyperbolic sets where they use the term "hyperbolic equilibrium point" instead of "hyperbolic point".