# The definition of C^r Structural Stability

I currently have a definition that states that given a flow $f$, $f$ is structurally stable if for any $g$ in some neighborhood of $f$, $f$ and $g$ are topologically conjugate.

Would the definition of $C^r$ Structural Stability be that for a flow$f$, $f$ is structurally stable if for any $g$ that is $C^r$ close to $f$ (i.e. $f$ and its first $r$ derivatives are within $\epsilon$ of $g$ and its first $r$ derivatives respectively), $f$ and $g$ are topologically conjugate?

I am a little confused on how this is a different definition than the first one. Any elaboration would be appreciated.

• The first definition coincides with the second one, if by "some neighborhood" you mean a neighborhood in the $C^r$ topology. – Hans Lundmark May 11 '15 at 20:01

Your definitions are right. The difference is that in the first case you have a flow (vector field) that may be only $C^1$-smooth and you are looking at its neighborhood in $C^1$-topology. For the second one your vector field should be at least $C^r$ and the neighbourhood in the $C^r$-topology.