Wandering set Totally Explained

Everything about Wandering Set totally explained

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.

Wandering points

A common, discrete-time definition of wandering sets starts with a map

f:X	o X

of a topological space X. A point

xin X

is said to be a wandering point if there's a neighbourhood U of x and a positive integer N such that for all

n>N

, the iterated map is non-intersecting: ť

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, for example part of a triple

(X,Sigma,mu)

of Borel sets

Sigma

and a measure

mu

such that ť

muleft(f^n(U) cap U 
ight) = 0,,

Similarly, a continuous-time system will have a map

varphi_t:X	o 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:
ť

varphi_ ;; 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 ť

Omega - W^*,

is a set of measure zero.

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