  Self-interactions and interaction with the environment tend to push quantum
systems toward states of maximal entanglement. This is a definition of
decoherence. We argue that these maximally entangled states fall into the
well-defined classes that can be uniquely described by the values of certain
entanglement invariants. After discussing these ideas we present examples of
maximally entangled states for a number of generic systems, construct compact
states in the most entangled classes for tripartite systems, and suggest how
they may be constructed for other $n$-partite systems. We study random walks
through the space of entanglement classes to see how decoherence might work in
practice.
