We develop several aspects of local and global stability in continuous first order logic. In particular, we study type-definable groups and genericity.
We prove that $IHS_A$, the theory of infinite dimensional Hilbert spaces equipped with a generic automorphism, is $aleph_0$-stable up to perturbation of the automorphism, and admits prime models up to perturbation over any set. Similarly, $APr_A$, th
e theory of atomless probability algebras equipped with a generic automorphism is $aleph_0$-stable up to perturbation. However, not allowing perturbation it is not even superstable.