# Wandering set

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{{no footnotes|date=June 2023}}
{{Short description|In mathematics, a concept that formalizes a certain idea of movement and mixing}}
In [dynamical system](/source/dynamical_system)s and [ergodic theory](/source/ergodic_theory), the concept of a '''wandering set''' formalizes a certain idea of movement and [mixing](/source/mixing_(mathematics)). When a dynamical system has a wandering set of non-zero measure, then the system is a [dissipative system](/source/dissipative_system). This is the opposite of a [conservative system](/source/conservative_system), to which the [Poincaré recurrence theorem](/source/Poincar%C3%A9_recurrence_theorem) applies. Intuitively, the connection between wandering sets and dissipation is easily understood: if a portion of the [phase space](/source/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](/source/George_David_Birkhoff) in 1927.{{citation needed|date=November 2010}}

==Wandering points==
A common, discrete-time definition of wandering sets starts with a map <math>f:X\to X</math> of a [topological space](/source/topological_space) ''X''. A point <math>x\in X</math> is said to be a '''wandering point''' if there is a [neighbourhood](/source/neighbourhood_(mathematics)) ''U'' of ''x'' and a positive integer ''N'' such that for all <math>n>N</math>, the [iterated map](/source/iterated_map) is non-intersecting:

:<math>f^n(U) \cap U = \varnothing.</math>

A handier definition requires only that the intersection have [measure zero](/source/measure_zero). To be precise, the definition requires that ''X'' be a [measure space](/source/measure_space), i.e. part of a triple <math>(X,\Sigma,\mu)</math> of [Borel set](/source/Borel_set)s <math>\Sigma</math> and a measure <math>\mu</math> such that

:<math>\mu\left(f^n(U) \cap U \right) = 0,</math>

for all <math>n>N</math>. Similarly, a continuous-time system will have a map <math>\varphi_t:X\to X</math> defining the time evolution or [flow](/source/flow_(mathematics)) of the system, with the time-evolution operator <math>\varphi</math> being a one-parameter continuous [abelian group](/source/abelian_group) [action](/source/Group_action_(mathematics)) on ''X'':

:<math>\varphi_{t+s} = \varphi_t \circ \varphi_s.</math>

In such a case, a wandering point <math>x\in X</math> will have a neighbourhood ''U'' of ''x'' and a time ''T'' such that for all times <math>t>T</math>, the time-evolved map is of measure zero:

:<math>\mu\left(\varphi_t(U) \cap U \right) = 0.</math>

These simpler definitions may be fully generalized to the [group action](/source/Group_action_(mathematics)) of a [topological group](/source/topological_group). Let <math>\Omega=(X,\Sigma,\mu)</math> be a measure space, that is, a [set](/source/set_(mathematics)) with a [measure](/source/measure_(mathematics)) defined on its [Borel subset](/source/Borel_subset)s.  Let <math>\Gamma</math>  be a group acting on that set. Given a point <math>x \in \Omega</math>, the set

:<math>\{\gamma \cdot x : \gamma \in \Gamma\}</math>

is called the [trajectory](/source/trajectory) or [orbit](/source/orbit_(group_theory)) of the point ''x''.

An element <math>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>\Gamma</math>  such that 
:<math>\mu\left(\gamma \cdot U \cap U\right)=0</math>

for all <math>\gamma \in \Gamma-V</math>.

==Non-wandering points==
A '''non-wandering point''' is the opposite. In the discrete case, <math>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>\mu\left(f^n(U)\cap U \right) > 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>\Omega</math> is a '''wandering set''' under the action of a discrete group <math>\Gamma</math> if ''W'' is measurable and if, for any <math>\gamma \in \Gamma - \{e\}</math> the intersection

:<math>\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>\Gamma</math> is said to be ''{{dfn|dissipative}}'', and the dynamical system <math>(\Omega, \Gamma)</math> is said to be a [dissipative system](/source/dissipative_system). If there is no such wandering set, the action is said to be ''{{dfn|conservative}}'', and the system is a [conservative system](/source/conservative_system). For example, any system for which the [Poincaré recurrence theorem](/source/Poincar%C3%A9_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>W^* = \bigcup_{\gamma \in \Gamma} \;\; \gamma W.</math>

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

:<math>\Omega - W^*</math>

is a set of measure zero.

The [Hopf decomposition](/source/Hopf_decomposition) states that every [measure space](/source/measure_space) with a [non-singular transformation](/source/conservative_system) can be decomposed into an invariant conservative set and an invariant wandering set.

==See also==
* [No wandering domain theorem](/source/No_wandering_domain_theorem)

==References==
* {{cite book |first=Peter J. |last=Nicholls |title=The Ergodic Theory of Discrete Groups |url=https://archive.org/details/ergodictheoryofd0000nich |url-access=registration |year=1989 |publisher=Cambridge University Press |location=Cambridge |isbn=0-521-37674-2 }}
* Alexandre I. Danilenko and Cesar E. Silva (8 April 2009). ''[https://web.williams.edu/Mathematics/csilva/NonsingularET_Apr.pdf Ergodic theory: Nonsingular transformations]''; See [https://arxiv.org/abs/0803.2424 Arxiv arXiv:0803.2424].
* {{citation|last=Krengel|first=Ulrich|title=Ergodic theorems|series=De Gruyter Studies in Mathematics|volume=6|publisher=de Gruyter|year= 1985|isbn=3-11-008478-3 }}

{{DEFAULTSORT:Wandering Set}}
Category:Ergodic theory
Category:Limit sets
Category:Dynamical systems

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Adapted from the Wikipedia article [Wandering set](https://en.wikipedia.org/wiki/Wandering_set) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Wandering_set?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.