Chaos and Lyapunov exponents in classical and quantal distribution dynamics

We analytically establish the role of a spectrum of Lyapunov exponents in the evolution of phase-space distributions $\rho \mathrm{}(p,q).$ Of particular interest is ${\lambda }_2,$ an exponent that quantifies the rate at which chaotically evolving distributions acquire structure at increasingly smaller scales and is generally larger than the maximal Lyapunov exponent λ for trajectories. The approach is trajectory independent and is therefore applicable to both classical and quantum mechanics. In the latter case we show that the $ħ\to 0$ limit yields the classical, fully chaotic, result for the quantum cat map.