Rigidity of symmetric products

Given a metric continuum X, we consider the following hyperspaces of X: 2 X , C n ( X ) and F n ( X ) ( n ∈ N ). Let F 1 ( X ) = { { x } : x ∈ X } . A hyperspace K ( X ) of X is said to be rigid provided that for every homeomorphism h : K ( X ) → K ( X ) we have that h ( F 1 ( X ) ) = F 1 ( X ) . In this paper we study under which conditions a continuum X has a rigid hyperspace F n ( X ) . 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.