In those branches of mathematics called dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and mixing in such systems. When a dynamical system has a wandering set of non-zero measure, then the system is a dissipative system. This is very much the opposite of a conservative system, for which the ideas of the Poincaré recurrence theorem apply. 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.
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Wandering points
A common, discrete-time definition of wandering sets starts with a map
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
Similarly, a continuous-time system will have a map
In such a case, a wandering point
These simpler definitions may be fully generalized to the group action of a topological group. Let
is called the trajectory or orbit of the point x.
An element
for all
Non-wandering points
A non-wandering point is the opposite. In the discrete case,
Similar definitions follow for the continuous-time and discrete and continuous group actions.
Wandering sets and dissipative systems
A wandering set is a collection of wandering points. More precisely, a subset W of
is a set of measure zero.
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
Define the trajectory of a wandering set W as
The action of
is a set of measure zero.