Dynamical forcing of circular groups
Abstract
Description
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.
18 pages (v3: typos corrected)
18 pages (v3: typos corrected)