A compact metric space that is universal for orbit spectra of homeomorphisms

We say a space X with property P is a universal space for orbit spectra of homeomorphisms with property P provided that if Y is any space with property P and the same cardinality as X and h:Y→Y is any (auto)homeomorphism then there is a homeomorphism g:X→X such that the orbit equivalence classes for h and g are isomorphic. We construct a compact metric space X that is universal for homeomorphisms of compact metric spaces of cardinality of the continuum c and prove that there is no such space that is countably infinite. In the presence of some set theoretic assumptions we also give a separable metric space of size c that is universal for homeomorphisms on separable metric spaces.