Our DMV Seminar on 'Classical Nonintegrability, Quantum Chaos' intended
to introduce students and beginning researchers to the techniques
applied in nonin- tegrable classical and quantum dynamics. Several of
these lectures are collected in this volume. The basic phenomenon of
nonlinear dynamics is mixing in phase space, lead- ing to a positive
dynamical entropy and a loss of information about the initial state. The
nonlinear motion in phase space gives rise to a linear action on phase
space functions which in the case of iterated maps is given by a
so-called transfer operator. Good mixing rates lead to a spectral gap
for this operator. Similar to the use made of the Riemann zeta function
in the investigation of the prime numbers, dynamical zeta functions are
now being applied in nonlinear dynamics. In Chapter 2 V. Baladi first
introduces dynamical zeta functions and transfer operators, illustrating
and motivating these notions with a simple one-dimensional dynamical
system. Then she presents a commented list of useful references, helping
the newcomer to enter smoothly into this fast-developing field of
research. Chapter 3 on irregular scattering and Chapter 4 on quantum
chaos by A. Knauf deal with solutions of the Hamilton and the
Schr6dinger equation. Scatter- ing by a potential force tends to be
irregular if three or more scattering centres are present, and a typical
phenomenon is the occurrence of a Cantor set of bounded orbits. The
presence of this set influences those scattering orbits which come
close.
