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.
A common, discrete-time definition of wandering sets starts with a map
f:X\toX
x\inX
n>N
fn(U)\capU=\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)
\Sigma
\mu
\mu\left(fn(U)\capU\right)=0,
for all
n>N
\varphit:X\toX
\varphi
\varphit+s=\varphit\circ\varphis.
In such a case, a wandering point
x\inX
t>T
\mu\left(\varphit(U)\capU\right)=0.
These simpler definitions may be fully generalized to the group action of a topological group. Let
\Omega=(X,\Sigma,\mu)
\Gamma
x\in\Omega
\{\gamma ⋅ x:\gamma\in\Gamma\}
is called the trajectory or orbit of the point x.
An element
x\in\Omega
\Gamma
\mu\left(\gamma ⋅ U\capU\right)=0
for all
\gamma\in\Gamma-V
A non-wandering point is the opposite. In the discrete case,
x\inX
\mu\left(fn(U)\capU\right)>0.
Similar definitions follow for the continuous-time and discrete and continuous group actions.
A wandering set is a collection of wandering points. More precisely, a subset W of
\Omega
\Gamma
\gamma\in\Gamma-\{e\}
\gammaW\capW
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
(\Omega,\Gamma)
Define the trajectory of a wandering set W as
W*=cup\gamma \gammaW.
The action of
\Gamma
W*
\Omega
\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.