Dynamical Systems of Algebraic Origin

Description

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. Front Matter ....Pages i-xviii Group actions by automorphisms of compact groups (Klaus Schmidt)....Pages 1-33 ℤd-actions on compact abelian groups (Klaus Schmidt)....Pages 35-75 Expansive automorphisms of compact groups (Klaus Schmidt)....Pages 77-92 Periodic points (Klaus Schmidt)....Pages 93-104 Entropy (Klaus Schmidt)....Pages 105-149 Positive entropy (Klaus Schmidt)....Pages 151-219 Zero entropy (Klaus Schmidt)....Pages 221-259 Mixing (Klaus Schmidt)....Pages 261-283 Rigidity (Klaus Schmidt)....Pages 285-300 Back Matter ....Pages 301-310