Structure of quantum chaotic wave functions: ergodicity, localization, and transport

We discuss recent developments in the study of quantum wave functions and transport in classically ergodic systems. Surprisingly, short-time classical dynamics leaves permanent imprints on long-time and stationary quantum behavior, which are absent from the long-time classical motion. These imprints can lead to quantum behavior on single-wavelength or single-channel scales that is very different from random matrix theory expectations. Robust and quantitative predictions are obtained using semiclassical methods. Applications to wave function intensity statistics and to resonances in open systems are discussed.