We study the set
D
n
,
β
(
R
d
)
\mathcal D^{n,\beta }(\mathbb R^d)
of orientation preserving diffeomorphisms of
R
d
\mathbb R^d
which differ from the identity by a Hölder
C
0
n
,
β
C^{n,\beta }_0
-mapping, where
n
∈
N
≥
1
n \in \mathbb N_{\ge 1}
and
β
∈
(
0
,
1
]
\beta \in (0,1]
. We show that
D
n
,
β
(
R
d
)
\mathcal D^{n,\beta }(\mathbb R^d)
forms a group, but left translations in
D
n
,
β
(
R
d
)
\mathcal D^{n,\beta }(\mathbb R^d)
are in general discontinuous. The groups
D
n
,
β
−
(
R
d
)
:=
⋂
α
>
β
D
n
,
α
(
R
d
)
\mathcal D^{n,\beta -}(\mathbb R^d) := \bigcap _{\alpha > \beta } \mathcal D^{n,\alpha }(\mathbb R^d)
(with its natural Fréchet topology) and
D
n
,
β
+
(
R
d
)
:=
⋃
α
>
β
D
n
,
α
(
R
d
)
\mathcal D^{n,\beta +}(\mathbb R^d) := \bigcup _{\alpha > \beta } \mathcal D^{n,\alpha }(\mathbb R^d)
(with its natural inductive locally convex topology) however are
C
0
,
ω
C^{0,\omega }
Lie groups for any slowly vanishing modulus of continuity
ω
\omega
. In particular,
D
n
,
β
−
(
R
d
)
\mathcal D^{n,\beta -}(\mathbb R^d)
is a topological group and a so-called half-Lie group (with smooth right translations). We prove that the Hölder spaces
C
0
n
,
β
C^{n,\beta }_0
are ODE closed, in the sense that pointwise time-dependent
C
0
n
,
β
C^{n,\beta }_0
-vector fields
u
u
have unique flows
Φ
\Phi
in
D
n
,
β
(
R
d
)
\mathcal D^{n,\beta }(\mathbb R^d)
. This includes, in particular, all Bochner integrable functions
u
∈
L
1
(
[
0
,
1
]
,
C
0
n
,
β
(
R
d
,
R
d
)
)
u \in L^1([0,1],C^{n,\beta }_0(\mathbb R^d,\mathbb R^d))
. For the latter and
n
≥
2
n\ge 2
, we show that the flow map
L
1
(
[
0
,
1
]
,
C
0
n
,
β
(
R
d
,
R
d
)
)
→
C
(
[
0
,
1
]
,
D
n
,
α
(
R
d
)
)
L^1([0,1],C^{n,\beta }_0(\mathbb R^d,\mathbb R^d)) \to C([0,1],\mathcal D^{n,\alpha }(\mathbb R^d))
,
u
↦
Φ
u \mapsto \Phi
, is continuous (even
C
0
,
β
−
α
C^{0,\beta -\alpha }
), for every
α
>
β
\alpha > \beta
. As an application we prove that the corresponding Trouvé group
G
n
,
β
(
R
d
)
\mathcal G_{n,\beta }(\mathbb R^d)
from image analysis coincides with the connected component of the identity of
D
n
,
β
(
R
d
)
\mathcal D^{n,\beta }(\mathbb R^d)
.