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].

Contents

1 Wandering points

2 Non-wandering points

3 Wandering sets and dissipative systems

4 See also

5 References

Wandering points[edit]

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

)