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.