  This paper began as a generalization of a part of the author's PhD thesis
about ACFA and ended up with a characterization of groups definable in T_A.
  The thesis concerns minimal formulae in ACFA of the form "p lies on an
algebraic curve A and s(x)=f(x)" for some dominant rational function f from A
to s(A), where s is the automorphism. These are shown to be uniform in the
Zilber trichotomy, and the pairs (A,f) that fall into each of the three cases
are characterized. These characterizations are definable in families.
  This paper covers approximately half of the thesis, namely those parts of it
which can be made purely model-theoretic by moving from ACFA, the model
companion of the class of algebraically closed fields with an endomorphism, to
T_A, the model companion of the class of models of an arbitrary
totally-transcendental theory T with an injective endomorphism, if this
model-companion exists.
  A T_A analog of the characterization of groups definable in ACFA is obtained
in the process. The full characterization is obtained from these intermediate
results with heavy use of algebraic geometry: see the thesis or the forthcoming
paper "Around Lattes functions".
