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Wikipedia / Wandering set

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.

Wandering points

of a topological space X is said to be a wandering point if there is a neighbourhood U of x and a positive integer N, 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 of Borel sets such that

defining the time evolution or flow being a one-parameter continuous abelian group action on X:

will have a neighbourhood U of x and a time T, the time-evolved map is of measure zero:

These simpler definitions may be fully generalized to the group action of a topological group be a measure space, that is, a set with a measure defined on its Borel subsets, the set

is called the trajectory or orbit of the point x.

is called a wandering point if there exists a neighborhood U of x and a neighborhood V such that

.

Non-wandering points

A non-wandering point 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 is a wandering set if W the intersection

is a set of measure zero.

is said to be 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

is said to be if there exists a wandering set W is almost-everywhere, 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

References