  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 of rotation numbers which can be
forced by finitely presented groups is an infinitely generated Q-module,
containing countably infinitely many algebraically independent transcendental
numbers.
  We also show that the set of subsets of the circle which are the set of
rotation numbers of an element g of a group G under all actions of G on a
circle, as G varies over all countable groups, are exactly the set of closed
subsets of the circle which contain 0, and are invariant under the involution
which interchanges x and -x.
  As another application, we construct a finitely generated group which acts
faithfully on the circle, but which does not admit any faithful C^1 action,
thus answering in the negative a question of John Franks.
