Wandering set

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


 * $$\{\gamma \cdot x : \gamma \in \Gamma\}$$

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
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


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

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.