Reversible dynamics

Mathematics
In mathematics, a dynamical system is invertible if the forward evolution is one-to-one, not many-to-one; so that for every state there exists a well-defined reverse-time evolution operator.
The dynamics are time-reversible if there exists a transformation (an involution) π which gives a one-to-one mapping between the time-reversed evolution of any one state, and the forward-time evolution of another corresponding state, given by the operator equation:
Any time-independent structures (for example critical points, or attractors) which the dynamics gives rise to must therefore either be self-symmetrical or have symmetrical images under the involution π.
Physics
In physics, the laws of motion of classical mechanics have the above property, if the operator π reverses the conjugate momenta of all the particles of the system, p -> -p . (T-symmetry).
In quantum...