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. The notion of wandering sets in phase space was introduced by Birkhoff in 1927[citation needed].

Wandering points[edit]

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.\,

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 \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[edit]

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 N and any N \ge 1 arbitrarily large. Similar definitions follow for the continuous-time and discrete and continuous group actions.

Wandering sets and dissipative systems[edit]

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^* = \cup_{\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.

See also[edit]

References[edit]

  • Nicholls, Peter J. (1989). The Ergodic Theory of Discrete Groups. Cambridge: Cambridge University Press. ISBN 0-521-37674-2.