A space that admits all possible orbit spectra of homeomorphisms of uncountable compact metric spaces

6 June 2012
Joint work with: Sina Greenwood, Brian Raines and Casey Sherman Abstract: We say a space $X$ with property $\C P$ is \emph{universal} for orbit spectra of homeomorphisms with property $\C P$ provided that if $Y$ is any space with property $\C P$ and the same cardinality as $X$ and $h:Y\to Y$ is any (auto)homeomorphism then there is a homeomorphism$g:X\to X$ 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 the continuum. There is no universal space for countable compact metric spaces. In the presence of some set theoretic assumptions we also give a separable metric space of size continuum that is universal for homeomorphisms on separable metric spaces.
  • Analytic Topology in Mathematics and Computer Science