Thompson's group may act by homeomorphisms on the circle.


Has this action a fixed point?



Orbits of Thompson's group
If you mean Thompson's $F$, please, specify the action on the circle.


If you mean Thompson's group $[F,F]$, the answer is yes, as it acts on the interval.


I think you are talking about Thompson's group $T$, and its dynamics on the circle was well described by Ghys and Sergiescu in


*[Sur un groupe remarquable de difféomorphismes du cercle.](http://www.umpa.ens-lyon.fr/~ghys/reviews/896095.pdf)*


The answer to your question is however rather trivial : the group contains every dyadic rotation, which act minimally on the circle. So every finite orbite must be dense, which is impossible.


