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

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

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

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

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

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

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

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

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

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

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

is called the trajectory or orbit of the point x.

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

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

for all $\gamma \in \Gamma -V$.

Non-wandering points

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

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

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

$\gamma W\cap W$

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

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

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

$\Omega -W^{*}$

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, vol. 6, de Gruyter, ISBN 3-11-008478-3