# Wandering set

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{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
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## Wandering points

A common, discrete-time definition of wandering sets starts with a map of a topological space *X*. A point 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 of Borel sets and a measure such that

Similarly, a continuous-time system will have a map 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 will have a neighbourhood *U* of *x* and a time *T* such that for all times , the time-evolved map is of measure zero:

These simpler definitions may be fully generalized to a general group action. Let be a measure space, that is, a set with a measure defined on its Borel subsets. Let be a group acting on that set. Given a point , the set

is called the trajectory or orbit of the point *x*.

An element is called a **wandering point** if there exists a neighborhood *U* of *x* and a neighborhood *V* of the identity in such that

## Non-wandering points

The definition for a **non-wandering point** is in a sense the converse. In the discrete case, is non-wandering if, for every open set *U* containing *x*, one has that

for some and any arbitrarily large. 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 **wandering set** under the action of a discrete group if *W* is measurable and if, for any the intersection

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 is said to be **dissipative**, and the dynamical system is said to be a dissipative system. If there is no such wandering set, the action is said to be **conservative**, and the system is a conservative system. 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 is almost-everywhere equal to , that is, if

is a set of measure zero.

## See also

## References

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