Although the study of dynamical systems is mainly concerned with single
trans- formations and one-parameter flows (i. e. with actions of Z, N,
JR, or JR+), er- godic theory inherits from statistical mechanics not
only its name, but also an obligation to analyze spatially extended
systems with multi-dimensional sym- metry groups. However, the wealth of
concrete and natural examples, which has contributed so much to the
appeal and development of classical dynamics, is noticeably absent in
this more general theory. A remarkable exception is provided by a class
of geometric actions of (discrete subgroups of) semi-simple Lie groups,
which have led to the discovery of one of the most striking new
phenomena in multi-dimensional ergodic theory: under suitable
circumstances orbit equivalence of such actions implies not only
measurable conjugacy, but the conjugating map itself has to be extremely
well behaved. Some of these rigidity properties are inherited by certain
abelian subgroups of these groups, but the very special nature of the
actions involved does not allow any general conjectures about actions of
multi-dimensional abelian groups. Beyond commuting group rotations,
commuting toral automorphisms and certain other algebraic examples (cf.
[39]) it is quite difficult to find non-trivial smooth Zd-actions on
finite-dimensional manifolds. In addition to scarcity, these examples
give rise to actions with zero entropy, since smooth Zd-actions with
positive entropy cannot exist on finite-dimensional, connected
manifolds. Cellular automata (i. e.