Starting with a unit-preserving normal completely positive map L: M --> M
acting on a von Neumann algebra - or more generally a dual operator system - we
show that there is a unique reversible system \alpha: N --> N (i.e., a complete
order automorphism \alpha of a dual operator system N) that captures all of the
asymptotic behavior of L, called the {\em asymptotic lift} of L. This provides
a noncommutative generalization of the Frobenius theorems that describe the
asymptotic behavior of the sequence of powers of a stochastic n x n matrix. In
cases where M is a von Neumann algebra, the asymptotic lift is shown to be a
W*-dynamical system (N,\mathbb Z), which we identify as the tail flow of the
minimal dilation of L. We also identify the Poisson boundary of L as the
fixed algebra of (N,\mathbb Z).
In general, we show the action of the asymptotic lift is trivial iff L is
{\em slowly oscillating} in the sense that $$ \lim_{n\to\infty}\|\rho\circ
L^{n+1}-\rho\circ L^n\|=0,\qquad \rho\in M_* . $$ Hence \alpha is often a
nontrivial automorphism of N.