Wandering set explained

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 ⋅ 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 ⋅ 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 , 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 , 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 ^{*} = c up _{\gamma} \gamma W.

The action of

\Gamma is said to be 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

• Book: Nicholls, Peter J. . The Ergodic Theory of Discrete Groups . registration . 1989 . Cambridge University Press . Cambridge . 0-521-37674-2 .
• Alexandre I. Danilenko and Cesar E. Silva (8 April 2009). Ergodic theory: Nonsingular transformations; See Arxiv arXiv:0803.2424.