Wandering set

Short description: In mathematics, a concept that formalizes a certain idea of movement and mixing

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

A common, discrete-time definition of wandering sets starts with a map [math]\displaystyle{ f:X\to X }[/math] of a topological space X. A point [math]\displaystyle{ x\in X }[/math] is said to be a wandering point if there is a neighbourhood U of x and a positive integer N such that for all [math]\displaystyle{ n\gt N }[/math], the iterated map is non-intersecting:

[math]\displaystyle{ f^n(U) \cap U = \varnothing. }[/math]

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 [math]\displaystyle{ (X,\Sigma,\mu) }[/math] of Borel sets [math]\displaystyle{ \Sigma }[/math] and a measure [math]\displaystyle{ \mu }[/math] such that

[math]\displaystyle{ \mu\left(f^n(U) \cap U \right) = 0, }[/math]

for all [math]\displaystyle{ n\gt N }[/math]. Similarly, a continuous-time system will have a map [math]\displaystyle{ \varphi_t:X\to X }[/math] defining the time evolution or flow of the system, with the time-evolution operator [math]\displaystyle{ \varphi }[/math] being a one-parameter continuous abelian group action on X:

[math]\displaystyle{ \varphi_{t+s} = \varphi_t \circ \varphi_s. }[/math]

In such a case, a wandering point [math]\displaystyle{ x\in X }[/math] will have a neighbourhood U of x and a time T such that for all times [math]\displaystyle{ t\gt T }[/math], the time-evolved map is of measure zero:

[math]\displaystyle{ \mu\left(\varphi_t(U) \cap U \right) = 0. }[/math]

These simpler definitions may be fully generalized to the group action of a topological group. Let [math]\displaystyle{ \Omega=(X,\Sigma,\mu) }[/math] be a measure space, that is, a set with a measure defined on its Borel subsets. Let [math]\displaystyle{ \Gamma }[/math] be a group acting on that set. Given a point [math]\displaystyle{ x \in \Omega }[/math], the set

[math]\displaystyle{ \{\gamma \cdot x : \gamma \in \Gamma\} }[/math]

is called the trajectory or orbit of the point x.

An element [math]\displaystyle{ x \in \Omega }[/math] is called a wandering point if there exists a neighborhood U of x and a neighborhood V of the identity in [math]\displaystyle{ \Gamma }[/math] such that

[math]\displaystyle{ \mu\left(\gamma \cdot U \cap U\right)=0 }[/math]

for all [math]\displaystyle{ \gamma \in \Gamma-V }[/math].

Non-wandering points

A non-wandering point is the opposite. In the discrete case, [math]\displaystyle{ x\in X }[/math] is non-wandering if, for every open set U containing x and every N > 0, there is some n > N such that

[math]\displaystyle{ \mu\left(f^n(U)\cap U \right) \gt 0. }[/math]

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 [math]\displaystyle{ \Omega }[/math] is a wandering set under the action of a discrete group [math]\displaystyle{ \Gamma }[/math] if W is measurable and if, for any [math]\displaystyle{ \gamma \in \Gamma - \{e\} }[/math] the intersection

[math]\displaystyle{ \gamma W \cap W }[/math]

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 [math]\displaystyle{ \Gamma }[/math] is said to be dissipative, and the dynamical system [math]\displaystyle{ (\Omega, \Gamma) }[/math] 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

[math]\displaystyle{ W^* = \bigcup_{\gamma \in \Gamma} \;\; \gamma W. }[/math]

The action of [math]\displaystyle{ \Gamma }[/math] is said to be completely dissipative if there exists a wandering set W of positive measure, such that the orbit [math]\displaystyle{ W^* }[/math] is almost-everywhere equal to [math]\displaystyle{ \Omega }[/math], that is, if

[math]\displaystyle{ \Omega - W^* }[/math]

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

• No wandering domain theorem

References

• Nicholls, Peter J. (1989). The Ergodic Theory of Discrete Groups. Cambridge: Cambridge University Press. ISBN 0-521-37674-2. https://archive.org/details/ergodictheoryofd0000nich. 
• Alexandre I. Danilenko and Cesar E. Silva (8 April 2009). Ergodic theory: Nonsingular transformations; See Arxiv arXiv:0803.2424.
• Krengel, Ulrich (1985), Ergodic theorems, De Gruyter Studies in Mathematics, 6, de Gruyter, ISBN 3-11-008478-3