Let $T$ be a homeomorphism from a topological space to itself with a set of cyclic points, for example on the 2-adic units $\Bbb Z_2^\times$ that $T$ cycles $(-1,1,\frac13,-\frac13)$.
In general, and in this specific case, must it cycle other points in the space?
It seems clear that in general it must not because e.g. in the discrete topology I think one could cycle the given points and leave all the others fixed and have a homeomorphism. But as one makes the topology coarser, there is less freedom. Is there some threshold where this forces other points to cycle?
I'm aware of one weaker statement, which is Sharkovski's theorem, which in the above case on $\Bbb R$ instead of $\Bbb Z_2^\times$ would guarantee cyclic points of orders $2$ as well as fixed points. Obviously cases can be manufactured, but are there broad categories of topologies that require the whole space cycles together?