# The space of diffeomorphisms on a manifold

It is well known that given a compact connected smooth manifold without boundary $M$, the set of diffeomorphisms $Diff^{r}(M)$ of $M$ for $r≥1$, is open in $C^{r}(M)$, the set of continuous functions (for example, Thm. 1.7 - Chap 2, in "Differential Topology", M. Hirsch).

My question is: does it follow that $Diff^{r+\alpha}(M) \subset C^{r+\alpha}(M)$ is open w.r.t. the $C^{r + \alpha}$ topology for $0< \alpha \leq 1$?

If it is so, I guess the same can be said about the subset of expanding maps, right? Anything about the subset of hyperbolic systems? I guess here things are strongly dependent on the manifold, along the lines of the work of Crovisier, Bonatti, etc.

Can someone give me some references? Thanks.

• Is the notation $C^{r+a}$ completely standard? What does it mean for you? – Dylan Thurston May 14 '14 at 12:39
• I mean $C^{r + Hölder}$. – user50774 May 14 '14 at 12:41
• JFYI: Many people would write instead of $C^{r+\alpha}$, the notation $C^{r,\alpha}$ for Holder spaces. – Willie Wong May 14 '14 at 14:13
• True. I think in Analysis the $C^{r, \alpha}$ is the standard notation, whereas in Dynamical Systems the $C^{r + \alpha}$ one. – user50774 May 14 '14 at 14:19

## 1 Answer

Yes, $Diff^{r,\alpha}(M)$ is open in $C^{r,\alpha}(M,M)$. You need a Riemannian metric on $M$ to even say what a $C^{r,\alpha}$-map $M\to M$ is. This is because the $C^1$-topology is coarser than the $C^{r,\alpha}$-topology: so the embedding from $C^{r,\alpha}(M,M)$ with the $C^{r,\alpha}$-topology to $C^{1}(M,M)$ with the $C^1$-topology is continuous, and the inverse image of the open set $Diff^1(M)$ is then the open set $Diff^{r,\alpha}(M)$. See

• Peter W. Michor: Manifolds of differentiable mappings. Shiva Mathematics Series 3, Shiva Publ., Orpington, (1980), scanned pdf

for more background.

• Thanks a lot @peter-michor ! The question came about because I am interested in perturbations of dynamical systems. The idea I initially had was to use the proof in the book of M. Hirsch, plus some interpolation inequalities given, for example, in Lectures of Elliptic and Parabilic Equations in Hoelder spaces, by Krylov. I didn't know your reference, so I've just downloaded your book and will take a close look at this. Kind regards. Christian Rodrigues – user50774 Jul 14 '14 at 12:07