Rigidity of symmetric products

Given a metric continuum X, we consider the following hyperspaces of X  : 2X2X, Cn(X)Cn(X) and Fn(X)Fn(X) (n∈Nn∈N). Let F1(X)={{x}:x∈X}F1(X)={{x}:x∈X}. A hyperspace K(X)K(X) of X   is said to be rigid provided that for every homeomorphism h:K(X)→K(X)h:K(X)→K(X) we have that h(F1(X))=F1(X)h(F1(X))=F1(X). In this paper we study under which conditions a continuum X   has a rigid hyperspace Fn(X)Fn(X).Among others, we consider families of continua such as, dendroids, Peano continua, indecomposable arc continua (all their proper nondegenerate subcontinua are arcs), hereditarily indecomposable continua and smooth fans.