Wandering set

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 nN, 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 nN. 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 tT, 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.

References

• Alexandre I. Danilenko and Cesar E. Silva (8 April 2009). Ergodic theory: Nonsingular transformations; See Arxiv arXiv:0803.2424.