Transitivity of orbits of maps symmetric under compact Lie groups (Q1331207)

Let \(\Gamma\) be a compact connected Lie group, \(M\) be a homogeneous \(\Gamma\)-space, \(I\) be a compact metric space and \(f : I \times M\) to itself be a continuous map equivariant under an orthogonal action of \(\Gamma\) on \(M\). It is shown that if the map \(f\) is chaotic in the sense that it is a mixing (sub)shift then generically \(f\) has orbits which are topologically transitive in their own group orbit. This result is related to the ergodic theory of cocycle extensions by compact Lie groups.