Homogeneity degree of hyperspaces of continua

A continuum is a compact connected metric space. Given a topological space X, for points p, q in X, we consider the equivalence relation ~ given by: p~q if and only if there exists a homeomorphism h of X to X such that h(p) = q. The number of classes of this relation is called the homogeneity degree of X and is denoted by hd(X). We consider the following hyperspaces of X:

Cn(X) = all subsets A of X which are  closed, nonempty and have at most n components, and Fn(X) =  all A in Cn(X)  with at most n points.

In this talk we will discuss the known results, old and new, about the degrees hd(Cn(X)) and hd(Fn(X)), where X is a continuum.