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

Direct limits of large orbits and the Knaster continuum homeomorphism group