Direct limits of large orbits and the Knaster continuum homeomorphism group

Abstract

The main result is that the group $\text{Homeo}(K)$ of homeomorphisms of the universal Knaster continuum contains an open subgroup with a comeager conjugacy class. Actually, this open subgroup is the very natural subgroup consisting of degree-one homeomorphisms. We give a general fact about finding comeager orbits in Polish group actions which are approximated densely by direct limits of actions with comeager orbits. The main theorem comes as a result of this fact and some finer analysis of the conjugacy action of the group ${\text{Homeo}}_{+}[0,1]$.

16 pages