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Wandering set

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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.

Wandering points

A common, discrete-time definition of wandering sets starts with a map f : X → X {\displaystyle f:X\to X} of a topological space X. A point x ∈ X {\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 n > N {\displaystyle n>N} , the iterated map is non-intersecting:

f n ( U ) ∩ U = ∅ . {\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 ( X , Σ , μ ) {\displaystyle (X,\Sigma ,\mu )} of Borel sets Σ {\displaystyle \Sigma } and a measure μ {\displaystyle \mu } such that

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

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

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

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

μ ( φ t ( U ) ∩ U ) = 0. {\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 Ω = ( X , Σ , μ ) {\displaystyle \Omega =(X,\Sigma ,\mu )} be a measure space, that is, a set with a measure defined on its Borel subsets. Let Γ {\displaystyle \Gamma } be a group acting on that set. Given a point x ∈ Ω {\displaystyle x\in \Omega } , the set

{ γ ⋅ x : γ ∈ Γ } {\displaystyle \{\gamma \cdot x:\gamma \in \Gamma \}}

is called the trajectory or orbit of the point x.

An element x ∈ Ω {\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 Γ {\displaystyle \Gamma } such that

μ ( γ ⋅ U ∩ U ) = 0 {\displaystyle \mu \left(\gamma \cdot U\cap U\right)=0}

for all γ ∈ Γ − V {\displaystyle \gamma \in \Gamma -V} .

Non-wandering points

A non-wandering point is the opposite. In the discrete case, x ∈ X {\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

μ ( f n ( U ) ∩ U ) > 0. {\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 Ω {\displaystyle \Omega } is a wandering set under the action of a discrete group Γ {\displaystyle \Gamma } if W is measurable and if, for any γ ∈ Γ − { e } {\displaystyle \gamma \in \Gamma -\{e\}} the intersection

γ W ∩ W {\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 Γ {\displaystyle \Gamma } is said to be dissipative, and the dynamical system ( Ω , Γ ) {\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

W ∗ = ⋃ γ ∈ Γ γ W . {\displaystyle W^{*}=\bigcup _{\gamma \in \Gamma }\;\;\gamma W.}

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

Ω − W ∗ {\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

References

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