Dynamics of expansive group actions on the circle
by Steven Hurder

ABSTRACT:

In this paper, we study the topological dynamics of $C^0$ and $C^1$-actions on the circle by a countably generated group $\G$, under the assumption that there is expansive orbit behavior. Our first result is a simple proof that if a $C^0$-action $\varphi$ is expansive, then $\G$ must have a free sub-semigroup on two generators, hence has exponential growth. For $C^1$-actions, we introduce the set of points $E(\varphi)$ which are infinitesimally expansive, and prove that the hyperbolic periodic points are dense in $E(\varphi)$. We show that if the set $E(\varphi)$ has an accumulation point in itself, then the geometric entropy $h(\varphi)$ must be positive. Finally, we prove that if $\K$ is a minimal set for a $C^1$-action $\varphi$, then either there is a $\G$-invariant probability measure supported on $\K$, or the hyperbolic periodic points are dense in $K$.