{{no footnotes|date=June 2023}} {{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.{{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 ''X''. A point <math>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>n>N</math>, the iterated map is non-intersecting:
:<math>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>(X,\Sigma,\mu)</math> of Borel sets <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 of the system, with the time-evolution operator <math>\varphi</math> being a one-parameter continuous abelian group action 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 of a topological group. Let <math>\Omega=(X,\Sigma,\mu)</math> be a measure space, that is, a set with a measure defined on its Borel subsets. 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 or orbit 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. If there is no such wandering set, the action is said to be ''{{dfn|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>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 equal to <math>\Omega</math>, that is, if
:<math>\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== * {{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