  For full-line Jacobi matrices, Schr\"odinger operators, and CMV matrices, we
show that being reflectionless, in the sense of the well-known property of
$m$-functions, is equivalent to a lack of reflection in the dynamics in the
sense that any state that goes entirely to $x=-\infty$ as $t\to -\infty $ goes
entirely to $x=\infty$ as $t\to\infty$. This allows us to settle a conjecture
of Deift and Simon from 1983 regarding ergodic Jacobi matrices.
