# Orbits of Thompson's group

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

Has this action a fixed point?

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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