Dynamical-Symmetry Breaking, Mechanism and CharacterizationBehaviour

The integrability of quantum systems is charactized by the algebraic notion of dynamical symmetry, hence the nonintegrability of quantum systems should be discussed topologically in reference to corresponding integrable ones. A topological approach is proposed. The topological map in general is shown to be constituted of the piecewise diffeomorphic maps interrupted by sudden nondiffeomorphic maps and thus has the similar typical feature of stretching-folding stretching ...... as the Smale horse-shoe map.