  Cartan's uniqueness theorem does not hold in general for CR mappings, but it
does hold under certain conditions guaranteeing extendibility of CR functions
to a fixed neighborhood. These conditions can be defined naturally for a wide
class of sets such as local real-analytic subvarieties or subanalytic sets, not
just submanifolds. Suppose that $V$ is a locally connected and locally closed
subset of ${\mathbb{C}}^n$ such that the hull constructed by contracting
analytic discs close to arbitrarily small neighborhoods of a point always
contains the point in the interior. Then restrictions of holomorphic functions
uniquely extend to a fixed neighborhood of the point. Using this extension, we
obtain a version of Cartan's uniqueness theorem for such sets. When $V$ is a
real-analytic subvariety, we can generalize the concept of infinitesimal CR
automorphism and also prove an analogue of the theorem. As an application of
these two results we show that, for circular subvarieties, the only
automorphisms, CR or infinitesimal, are linear.
