Measuring the Lyapunov exponent using quantum mechanics.

We study the time evolution of two wave packets prepared at the same initial state, but evolving under slightly different Hamiltonians. For chaotic systems, we determine the circumstances that lead to an exponential decay with time of the wave packet overlap function. We show that for sufficiently weak perturbations, the exponential decay follows a Fermi golden rule, while by making the difference between the two Hamiltonians larger, the characteristic exponential decay time becomes the Lyapunov exponent of the classical system. We illustrate our theoretical findings by investigating numerically the overlap decay function of a two-dimensional dynamical system.

Specific heat of multifractal energy spectra.

Motivated by the self-similar character of energy spectra demonstrated for quasicrystals, we investigate the case of multifractal energy spectra, and compute the specific heat associated with simple archetypal forms of multifractal sets as generated by iterated maps. We considered the logistic map and the circle map at their threshold to chaos. Both examples show nontrivial structures associated with the scaling properties of their respective chaotic attractors. The specific heat displays generically log-periodic oscillations around a value that characterizes a single exponent, the "fractal dimension," of the distribution of energy levels close to the minimum value set to 0. It is shown that when the fractal dimension and the frequency of log oscillations of the density of states are large, the amplitude of the resulting log oscillation in the specific heat becomes much smaller than the log-periodic oscillation measured on the density of states.

The quantized D-transformation.

We construct a new example of a quantum map, the quantized version of the D-transformation, which is the natural extension to two dimensions of the tent map. The classical, quantum and semiclassical behavior is studied. We also exhibit some relationships between the quantum versions of the D-map and the parity projected baker's map. The method of construction allows a generalization to dissipative maps which includes the quantization of a horseshoe. (c) 1996 American Institute of Physics.