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.
Wandering points.
A common, discrete-time definition of wandering sets starts with a map formula_1 of a topological space "X". A point formula_2 is said to be a wandering point if there is a neighbourhood "U" of "x" and a positive integer "N" such that for all formula_3, 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 formula_5 of Borel sets formula_6 and a measure formula_7 such that
for all formula_3. Similarly, a continuous-time system will have a map formula_10 defining the time evolution or flow of the system, with the time-evolution operator formula_11 being a one-parameter continuous abelian group action on "X":
In such a case, a wandering point formula_2 will have a neighbourhood "U" of "x" and a time "T" such that for all times formula_14, the time-evolved map is of measure zero:
These simpler definitions may be fully generalized to the group action of a topological group. Let formula_16 be a measure space, that is, a set with a measure defined on its Borel subsets. Let formula_17 be a group acting on that set. Given a point formula_18, the set
is called the trajectory or orbit of the point "x".
An element formula_18 is called a wandering point if there exists a neighborhood "U" of "x" and a neighborhood "V" of the identity in formula_17 such that 
for all formula_23.
Non-wandering points.
A non-wandering point is the opposite. In the discrete case, formula_2 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 formula_26 is a wandering set under the action of a discrete group formula_17 if "W" is measurable and if, for any formula_28 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 formula_17 is said to be ', and the dynamical system formula_31 is said to be a dissipative system. If there is no such wandering set, the action is said to be ', 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 formula_17 is said to be "" if there exists a wandering set "W" of positive measure, such that the orbit formula_34 is almost-everywhere equal to formula_26, 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.