This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (June 2023) (Learn how and when to remove this message)

In dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and mixing. When a dynamical system has a wandering set of non-zero measure, then the system is a dissipative system. 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]

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

for all . 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 the group action of a topological group. 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

for all .

Non-wandering points

A non-wandering point is the opposite. In the discrete case, 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.

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.

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.

See also