In dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and mixing.
This is the opposite of a conservative system, to which the Poincaré recurrence theorem applies.
Intuitively, the connection between wandering sets and dissipation is easily understood: if a portion of the phase space "wanders away" during normal time-evolution of the system, and is never visited again, then the system is dissipative.
The language of wandering sets can be used to give a precise, mathematical definition to the concept of a dissipative system.
The notion of wandering sets in phase space was introduced by Birkhoff in 1927.
[citation needed] A common, discrete-time definition of wandering sets starts with a map
is said to be a wandering point if there is a neighbourhood U of x and a positive integer N such that for all
, the iterated map is non-intersecting: A handier definition requires only that the intersection have measure zero.
To be precise, the definition requires that X be a measure space, i.e. part of a triple
defining the time evolution or flow of the system, with the time-evolution operator
being a one-parameter continuous abelian group action on X: In such a case, a wandering point
, the time-evolved map is of measure zero: These simpler definitions may be fully generalized to the group action of a topological group.
, the set is called the trajectory or orbit of the point x.
is non-wandering if, for every open set U containing x and every N > 0, there is some n > N such that Similar definitions follow for the continuous-time and discrete and continuous group actions.
is a wandering set under the action of a discrete group
The concept of a wandering set is in a sense dual to the ideas expressed in the Poincaré recurrence theorem.
If there exists a wandering set of positive measure, then the action of
For example, any system for which the Poincaré recurrence theorem holds cannot have, by definition, a wandering set of positive measure; and is thus an example of a conservative system.
Define the trajectory of a wandering set W as The action of
is said to be completely dissipative if there exists a wandering set W of positive measure, such that the orbit
The Hopf decomposition states that every measure space with a non-singular transformation can be decomposed into an invariant conservative set and an invariant wandering set.