Endpoint-homogeneous fans

A fan $F$ is endpoint-homogeneous if for any two endpoints $e,e’$ of $F$, there is a homeomorphism $h: F \to F$ such that $h(e) = e’$.

We prove there are uncountably many distinct homeomorphism types of endpoint-homogeneous smooth fans. To do this, we associate to each such fan $F$ a topological invariant, in the form of an equivalence class $\mathrm{EPG}(F)$ of subsets of $[0,1]$ describing how the endpoints of $F$ limit onto any given blade of $F$. We describe precisely all the uncountably many different classes that can arise as $\mathrm{EPG}(F)$ for some endpoint-homogeneous smooth fan $F$. We also prove the existence of $\frac{1}{n}$-homogeneous smooth fans for all $n \geq 5$.