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.



The space of diffeomorphisms on a manifold
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](http://www.mat.univie.ac.at/~michor/manifolds_of_differentiable_mappings.pdf)


for more background.


