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We show how some basic dynamical ideas can be brought to bear on the study of Cartan geometries. In our main results, we give conditions for certain types of Cartan geometries to have constant curvature. We also consider ergodic actions of `higher-rank' semisimple groups on bundles supporting (not necessarily invariant) Cartan connections and show that these are `standard locally homogeneous' actions provided that some noncompact 1-parameter subgroup `preserves' a Cartan geometry.