Dynamics of expansive group actions on the circle
by Steven Hurder

HISTORY:

This paper had three sources. First, it is a write up of a quicker and more general proof that given an expansive action of a group G on the circle, then G must have exponential growth. This proof was described in a talk at the Newton Institute, Cambridge, UK on June 30, 2000.

The paper also gives proof for the circle of several of the theorems in the paper "Entropy and Dynamics of $C^1$ Foliations". The proofs for a group action on the circle are much simpler than in the foliation setting, so the ideas are more transparent.

Finally, the paper shows how to use an idea in a paper by Benson Farb and Peter Shalen to reduce a conjecture of Ghys about group actions on the circle to a question of existence of maps with isolated fixed-points on a minimal set, and includes the application of this reduction to prove the conjecture for real analytic foliatioons. The observation that the Farb-Shalen paper could be applied here was made in April, 2000 when writing up the first version of the proof of the "expansive actions have exponential growth" result. Conversations with Etienne Ghys and Gregory Margulis in July, 2000 about the conjecture suggested that it was useful to have this remark in written form.

The paper was posted on Monday, August 28, 2000. The following Thursday, Margulis circulated a preprint dated August 30 giving a proof of the Ghys Conjecture: An action by homeomorphisms of a group G either has an invariant measure, or there is a nonabelian free subgroup in G.

The manuscript was extensively revised and reposted on May 29, 2005, and includes an additional result: for a C^1 action, there is either an invariant measure, or the hyperbolic set of the action is non-empty. This result is based on a new cocycle tempering technique.