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 $\rho (p,q)$. Of particular interest is ${\lambda }_2$, an exponent which quantifies the rate at which chaotically evolving distributions acquire structure at increasingly smaller scales and which is generally larger than the maximal Lyapunov exponent $\lambda$ 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 $\mathrm{\hslash }\to 0$ limit yields the classical, fully chaotic, result for the quantum cat map.