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}.