In this paper we introduce and study the notion of dynamical forcing. Basically, we develop a toolkit of techniques to produce finitely presented groups which can only act on the circle with certain prescribed dynamical properties. As an application, we show that the set
X
⊂
R
/
Z
X \subset \mathbb {R}/\mathbb {Z}
consisting of rotation numbers
θ
\theta
which can be forced by finitely presented groups is an infinitely generated
Q
\mathbb {Q}
–module, containing countably infinitely many algebraically independent transcendental numbers. Here a rotation number
θ
\theta
is forced by a pair
(
G
θ
,
α
)
(G_\theta ,\alpha )
, where
G
θ
G_\theta
is a finitely presented group
G
θ
G_\theta
and
α
∈
G
θ
\alpha \in G_\theta
is some element, if the set of rotation numbers of
ρ
(
α
)
\rho (\alpha )
as
ρ
\rho
varies over
ρ
∈
Hom
(
G
θ
,
Homeo
+
(
S
1
)
)
\rho \in \operatorname {Hom}(G_\theta ,\operatorname {Homeo}^+(S^1))
is precisely the set
{
0
,
±
θ
}
\lbrace 0, \pm \theta \rbrace
. We show that the set of subsets of
R
/
Z
\mathbb {R}/\mathbb {Z}
which are of the form
\[
rot
(
X
(
G
,
α
)
)
=
{
r
∈
R
/
Z
|
r
=
rot
(
ρ
(
α
)
)
,
ρ
∈
Hom
(
G
,
Homeo
+
(
S
1
)
)
}
,
\operatorname {rot}(X(G,\alpha )) = \lbrace r \in \mathbb {R}/\mathbb {Z} \; | \; r = \operatorname {rot}(\rho (\alpha )), \rho \in \operatorname {Hom}(G,\operatorname {Homeo}^+(S^1)) \rbrace ,
\]
where
G
G
varies over countable groups, are exactly the set of closed subsets which contain
0
0
and are invariant under
x
→
−
x
x \to -x
. Moreover, we show that every such subset can be approximated from above by
rot
(
X
(
G
i
,
α
i
)
)
\operatorname {rot}(X(G_i,\alpha _i))
for finitely presented
G
i
G_i
. As another application, we construct a finitely generated group
Γ
\Gamma
which acts faithfully on the circle, but which does not admit any faithful
C
1
C^1
action, thus answering in the negative a question of John Franks.