  For the purpose of determining global properties of Cartan geometries from
local information about automorphisms, we introduce a construction for a Cartan
geometry that captures the local behavior of a given automorphism near a
distinguished element. The result of this construction, which we call the
sprawl generated by the automorphism from the distinguished element, is
uniquely characterized by a kind of "universal property" that allows us to
compare Cartan geometries admitting automorphisms with equivalent local
behavior. As example applications, we prove that the only affine structure on a
connected manifold admitting an affine transformation with scaling isotropy is
the standard one on affine space, and describe how to construct non-flat real
projective structures admitting nontrivial automorphisms with higher-order
fixed points.
