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 ${\displaystyle f:X\to X}$ of a topological space X. A point ${\displaystyle x\in X}$ is said to be a wandering point if there is a neighbourhood U of x and a positive integer N such that for all ${\displaystyle n>N}$, the iterated map is non-intersecting:

${\displaystyle f^{n}(U)\cap U=\varnothing .}$

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 ${\displaystyle (X,\Sigma ,\mu )}$ of Borel sets ${\displaystyle \Sigma }$ and a measure ${\displaystyle \mu }$ such that

${\displaystyle \mu \left(f^{n}(U)\cap U\right)=0,}$

for all ${\displaystyle n>N}$. Similarly, a continuous-time system will have a map ${\displaystyle \varphi _{t}:X\to X}$ defining the time evolution or flow of the system, with the time-evolution operator ${\displaystyle \varphi }$ being a one-parameter continuous abelian group action on X:

${\displaystyle \varphi _{t+s}=\varphi _{t}\circ \varphi _{s}.}$

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

${\displaystyle \mu \left(\varphi _{t}(U)\cap U\right)=0.}$

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

${\displaystyle \{\gamma \cdot x:\gamma \in \Gamma \}}$

is called the trajectory or orbit of the point x.

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

${\displaystyle \mu \left(\gamma \cdot U\cap U\right)=0}$

for all ${\displaystyle \gamma \in \Gamma -V}$.

Non-wandering points

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

${\displaystyle \mu \left(f^{n}(U)\cap U\right)>0.}$

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

${\displaystyle \gamma W\cap W}$

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 ${\displaystyle \Gamma }$ is said to be dissipative, and the dynamical system ${\displaystyle (\Omega ,\Gamma )}$ 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

${\displaystyle W^{*}=\bigcup _{\gamma \in \Gamma }\;\;\gamma W.}$

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

${\displaystyle \Omega -W^{*}}$

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.
• 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