Rigid sets in the Hilbert cube (Q1072141)

An embedding of a compact metric space X into the Hilbert cube is said to be rigid provided the only homeomorphism of X to itself that extends to a homeomorphism of the Hilbert cube is the identity homeomorphism. Evidently, a space C that has an isolated point cannot have a rigid embedding into the Hilbert cube. The authors show that this is the only ''obstruction'' by showing that if a compact metric space X has no isolated points, then X has uncountably many distinct rigid embeddings into the Hilbert cube. This work extends earlier work, in the finite dimensional setting, of the second author and, as that work was, is based on an analysis of ''essentially'' distinct wild embeddings of the Cantor set.