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

作者：Good Chris; Greenwood Sina; Raines Brian E^*; Sherman Casey

来源：Advances in Mathematics, 2012, 229(5): 2670-2685.

DOI：10.1016/j.aim.2012.01.001

• Abstract dynamical system
• Topological dynamical system
• Universal space
• Cycle type
• Orbit spectrum
• Homeomorphisms of separable metric spaces
• Homeomorphisms of compact metric spaces

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 -%26gt; Y is any (auto)homeomorphism then there is a homeomorphism g : X -%26gt; 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.

• 出版日期2012-3-20