Wandering set – Wikipedia

Posted on March 6, 2022 by lordneo

In mathematics, a concept that formalizes a certain idea of movement and mixing

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.^{[citation needed]}

Table of Contents

Wandering points[edit]

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

{displaystyle f:Xto X}

$f:Xto X$ of a topological space X. A point

{displaystyle xin X}

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

{displaystyle n>N}

$n>N”/></span>, the iterated map is non-intersecting: </p><div class='code-block code-block-4' style='margin: 8px 0; clear: both;'> <amp-ad width=$

{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

{displaystyle (X,Sigma ,mu )}

$(X,Sigma,mu)$ of Borel sets

{displaystyle Sigma }

$Sigma$ and a measure

{displaystyle mu }

$mu$ such that

{displaystyle mu left(f^{n}(U)cap Uright)=0,}

for all

{displaystyle n>N}

$n>N”/></span>. Similarly, a continuous-time system will have a map <span class=$

{displaystyle varphi _{t}:Xto X}

$varphi _{t}:Xto X$ defining the time evolution or flow of the system, with the time-evolution operator

{displaystyle varphi }

$varphi$ being a one-parameter continuous abelian group action on X:

{displaystyle varphi _{t+s}=varphi _{t}circ varphi _{s}.}

In such a case, a wandering point

{displaystyle xin X}

$xin X$ will have a neighbourhood U of x and a time T such that for all times

{displaystyle t>T}

$t>T”/></span>, the time-evolved map is of measure zero: </p> <dl> <dd><span class=$

{displaystyle mu left(varphi _{t}(U)cap Uright)=0.}

${displaystyle mu left(varphi _{t}(U)cap Uright)=0.}$

These simpler definitions may be fully generalized to the group action of a topological group. Let

{displaystyle Omega =(X,Sigma ,mu )}

$Omega =(X,Sigma ,mu )$ be a measure space, that is, a set with a measure defined on its Borel subsets. Let

{displaystyle Gamma }

$Gamma$ be a group acting on that set. Given a point

{displaystyle xin Omega }

$x in Omega$ , the set

{displaystyle {gamma cdot x:gamma in Gamma }}

is called the trajectory or orbit of the point x.

An element

{displaystyle xin Omega }

$x in Omega$ is called a wandering point if there exists a neighborhood U of x and a neighborhood V of the identity in

{displaystyle Gamma }

$Gamma$ such that

{displaystyle mu left(gamma cdot Ucap Uright)=0}

for all

{displaystyle gamma in Gamma -V}

$gamma in Gamma -V$ .

Non-wandering points[edit]

A non-wandering point is the opposite. In the discrete case,

{displaystyle xin X}

$xin X$ is non-wandering if, for every open set U containing x and every N > 0, there is some n > N such that

{displaystyle mu left(f^{n}(U)cap Uright)>0.}