For a fixed C*-algebra A, we consider all noncommutative
dynamical systems that can be generated by A. More
precisely, an A-dynamical system is a triple (i,B,\alpha)
where $\alpha$ is a *-endomorphism of a C*-algebra B,
and i: A-->B is the inclusion of A as a
C*-subalgebra with the property that B is
generated by A\cup \alpha(A)\cup \alpha^2(A)\cup....
There is a natural hierarchy in the class of A-dynamical
systems, and there is a universal one that dominates
all others, denoted (i,PA,\alpha).
We establish certain properties of (i,PA,\alpha)
and give applications to some concrete issues of
noncommutative dynamics.
For example, we show that every contractive completely
positive linear map \phi: A --> A gives rise to
to a unique A-dynamical system (i,B,\alpha) that
is ``minimal" with respect to \phi, and
we show that its C*-algebra B
can be embedded in the multiplier
algebra of A\otimes {\mathcal K}.