A Cantor set is a homeomorphic image of the standard ternary Cantor set $T$. Suppose that we have a Cantor set $C$ on the plane. It is well know that $C$ is in fact ambiently homeomorphic to $T$, meaning that the homeomorphism $h:C\to T$ extends to a homeomorphism of the entire plane $\mathbb{R}^2$.
My question is: Suppose that $E$ is the set of endpoints of the removed intervals in the construction of $T$. Is there a property that points in $C$ must satisfy so that they get mapped to points in $E$ under $h$? In other words given a point in $C$ can someone tell, before knowing where it gets mapped under $h$, whether it will get mapped to an endpoint or not?
If the answer is no in general is there any property we could impose on any point in $C$ so that it would be impossible for it to get mapped to a point in $E$ under $h$?
Added later: For example if the Cantor set $T$ lies on the real line then a point of $T$ that can be approximated by both sides from points in $T$ cannot get mapped by $h$ to a point in $E$ since those points can be approximated only by one side by points in $C$.
The Cantor set is countable dense homogeneous, so we can map the endpoints of the ternary Cantor set to any countable dense subset of a Cantor set.