Involutions of Hilbert cubes that are hyperspaces of Peano continua

Let α be an involution of a Peano continuum X with nowhere dense fixed point set. Let α⁎ be the induced involution on the hyperspace 2X of nonempty closed subsets of X topologized by a Hausdorff metric. Let E⊆2X be a non-degenerate, α⁎-invariant hyperspace of X that is an inclusion or growth hyperspace in the sense of Curtis and Schori, and assume that the complement of {X} in E is contractible. Let S(E) be its fixed point set. If E is an inclusion hyperspace, then the restriction αˆ⁎ of α⁎ to E is conjugate with the involution id×τ of the Hilbert cube S(E)×Πi≥1Ii, where τ is the involution of Πi≥1Ii that reflects each coordinate across its mid-point. If E is a growth hyperspace of X and X contains no open subset homeomorphic to an arc, then the same result holds. In either case, if the complement of {X} in S(E) is contractible, then αˆ⁎ is conjugate with the involution σ of Πi≥1Ii that reflects each even coordinate across its mid-point.