Wandering set

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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.

1 Wandering points
2 Non-wandering points
3 Wandering sets and dissipative systems
4 See also
5 References

[edit] 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 $(X,Σ,μ)$ of Borel sets $Σ$ and a measure $μ$ such that

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

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 a general group action. Let $Ω = (X,Σ,μ)$ be a measure space, that is, a set with a measure defined on its Borel subsets. Let $Γ$ 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 $Γ$ such that

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

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

[edit] Non-wandering points

The definition for a non-wandering point is in a sense the converse. In the discrete case, $x\in X$ is non-wandering if, for every open set U containing x, one has that

$\mu\left(f^n(U)\cap U \right) > 0\,$

for some $n\ge 1$ . Similar definitions follow for the continuous-time and discrete and continuous group actions.

[edit] Wandering sets and dissipative systems

A wandering set is a collection of wandering points. More precisely, a subset W of $Ω$ is a wandering set under the action of a discrete group $Γ$ 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 $Γ$ is said to be dissipative, and the dynamical system $(Ω,Γ)$ 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 are thus examples of conservative systems.

Define the trajectory of a wandering set W as

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

The action of $Γ$ 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 $Ω$ , that is, if

$\Omega - W^*\,$