Wandering set

<h2 id="wandering-points">Wandering points</h2>
A common, discrete-time definition of wandering sets starts with a map 
 
 
 
 f
 :
 X
 →
 X
 
 
 {\displaystyle f:X\to X}
 
 of a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> X. A point 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
 is said to be a wandering point if there is a <a href="/facts/Neighbourhood_(mathematics)/XHKgrpkp">neighbourhood</a> U of x and a positive integer N such that for all 
 
 
 
 n
 >
 N
 
 
 {\displaystyle n>N}
 
, the <a href="/facts/Iterated_map/Vzsx64An">iterated map</a> is non-intersecting:

f
          
            n
          
        
        (
        U
        )
        ∩
        U
        =
        ∅
        .
      
    
    {\displaystyle f^{n}(U)\cap U=\varnothing .}

A handier definition requires only that the intersection have <a href="/facts/Measure_zero/kMudL8Mn">measure zero</a>. To be precise, the definition requires that X be a <a href="/facts/Measure_space/w9Dx4KY1">measure space</a>, i.e. part of a triple 
 
 
 
 (
 X
 ,
 Σ
 ,
 μ
 )
 
 
 {\displaystyle (X,\Sigma ,\mu )}
 
 of <a href="/facts/Borel_set/jvSsfpkP">Borel sets</a> 
 
 
 
 Σ
 
 
 {\displaystyle \Sigma }
 
 and a measure 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 such that

μ
        
          (
          
            
              f
              
                n
              
            
            (
            U
            )
            ∩
            U
          
          )
        
        =
        0
        ,
      
    
    {\displaystyle \mu \left(f^{n}(U)\cap U\right)=0,}

for all 
 
 
 
 n
 >
 N
 
 
 {\displaystyle n>N}
 
. Similarly, a continuous-time system will have a map 
 
 
 
 
 φ
 
 t
 
 
 :
 X
 →
 X
 
 
 {\displaystyle \varphi _{t}:X\to X}
 
 defining the time evolution or <a href="/facts/Flow_(mathematics)/CrmJPfeV">flow</a> of the system, with the time-evolution operator 
 
 
 
 φ
 
 
 {\displaystyle \varphi }
 
 being a one-parameter continuous <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a> <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">action</a> on X:

φ
          
            t
            +
            s
          
        
        =
        
          φ
          
            t
          
        
        ∘
        
          φ
          
            s
          
        
        .
      
    
    {\displaystyle \varphi _{t+s}=\varphi _{t}\circ \varphi _{s}.}

In such a case, a wandering point 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
 will have a neighbourhood U of x and a time T such that for all times 
 
 
 
 t
 >
 T
 
 
 {\displaystyle t>T}
 
, the time-evolved map is of measure zero:

μ
        
          (
          
            
              φ
              
                t
              
            
            (
            U
            )
            ∩
            U
          
          )
        
        =
        0.
      
    
    {\displaystyle \mu \left(\varphi _{t}(U)\cap U\right)=0.}

These simpler definitions may be fully generalized to the <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">group action</a> of a <a href="/facts/Topological_group/cAgNMMRU">topological group</a>. Let 
 
 
 
 Ω
 =
 (
 X
 ,
 Σ
 ,
 μ
 )
 
 
 {\displaystyle \Omega =(X,\Sigma ,\mu )}
 
 be a measure space, that is, a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> with a <a href="/facts/Measure_(mathematics)/yCq7zlI4">measure</a> defined on its <a href="/facts/Borel_subset/jvSsfpkP">Borel subsets</a>. Let 
 
 
 
 Γ
 
 
 {\displaystyle \Gamma }
 
 be a group acting on that set. Given a point 
 
 
 
 x
 ∈
 Ω
 
 
 {\displaystyle x\in \Omega }
 
, the set

{
        γ
        ⋅
        x
        :
        γ
        ∈
        Γ
        }
      
    
    {\displaystyle \{\gamma \cdot x:\gamma \in \Gamma \}}

is called the <a href="/facts/Trajectory/STNN33FK">trajectory</a> or <a href="/facts/Orbit_(group_theory)/Vh9VtUUw">orbit</a> of the point x.
An element 
 
 
 
 x
 ∈
 Ω
 
 
 {\displaystyle x\in \Omega }
 
 is called a wandering point if there exists a neighborhood U of x and a neighborhood V of the identity in 
 
 
 
 Γ
 
 
 {\displaystyle \Gamma }
 
 such that

μ
        
          (
          
            γ
            ⋅
            U
            ∩
            U
          
          )
        
        =
        0
      
    
    {\displaystyle \mu \left(\gamma \cdot U\cap U\right)=0}

for all 
 
 
 
 γ
 ∈
 Γ
 −
 V
 
 
 {\displaystyle \gamma \in \Gamma -V}
 
.

<h2 id="non-wandering-points">Non-wandering points</h2>
A non-wandering point is the opposite. In the discrete case, 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle 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

μ
        
          (
          
            
              f
              
                n
              
            
            (
            U
            )
            ∩
            U
          
          )
        
        >
        0.
      
    
    {\displaystyle \mu \left(f^{n}(U)\cap U\right)>0.}

Similar definitions follow for the continuous-time and discrete and continuous group actions.

<h2 id="wandering-sets-and-dissipative-systems">Wandering sets and dissipative systems</h2>
A wandering set is a collection of wandering points. More precisely, a subset W of 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
 is a wandering set under the action of a discrete group 
 
 
 
 Γ
 
 
 {\displaystyle \Gamma }
 
 if W is measurable and if, for any 
 
 
 
 γ
 ∈
 Γ
 −
 {
 e
 }
 
 
 {\displaystyle \gamma \in \Gamma -\{e\}}
 
 the intersection

γ
        W
        ∩
        W
      
    
    {\displaystyle \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 
 
 
 
 Γ
 
 
 {\displaystyle \Gamma }
 
 is said to be dissipative, and the dynamical system 
 
 
 
 (
 Ω
 ,
 Γ
 )
 
 
 {\displaystyle (\Omega ,\Gamma )}
 
 is said to be a <a href="/facts/Dissipative_system/aTAhXfo8">dissipative system</a>. If there is no such wandering set, the action is said to be conservative, and the system is a <a href="/facts/Conservative_system/cJvgzFtb">conservative system</a>. For example, any system for which the <a href="/facts/Poincar%25C3%25A9_recurrence_theorem/Uz2ja34X">Poincaré recurrence theorem</a> 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
          
            ∗
          
        
        =
        
          ⋃
          
            γ
            ∈
            Γ
          
        
        
        
        γ
        W
        .
      
    
    {\displaystyle W^{*}=\bigcup _{\gamma \in \Gamma }\;\;\gamma W.}

The action of 
 
 
 
 Γ
 
 
 {\displaystyle \Gamma }
 
 is said to be completely dissipative if there exists a wandering set W of positive measure, such that the orbit 
 
 
 
 
 W
 
 ∗
 
 
 
 
 {\displaystyle W^{*}}
 
 is <a href="/facts/Almost-everywhere/1quRFtJE">almost-everywhere</a> equal to 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
, that is, if

Ω
        −
        
          W
          
            ∗
          
        
      
    
    {\displaystyle \Omega -W^{*}}

is a set of measure zero.
The <a href="/facts/Hopf_decomposition/b7VAPYII">Hopf decomposition</a> states that every <a href="/facts/Measure_space/w9Dx4KY1">measure space</a> with a <a href="/facts/Conservative_system/cJvgzFtb">non-singular transformation</a> can be decomposed into an invariant conservative set and an invariant wandering set.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/No_wandering_domain_theorem/uuMth8xv">No wandering domain theorem</a></li></ul>

<ul><li>Nicholls, Peter J. (1989). <a href="https://archive.org/details/ergodictheoryofd0000nich">The Ergodic Theory of Discrete Groups</a>. Cambridge: Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-37674-2.</li>
<li>Alexandre I. Danilenko and Cesar E. Silva (8 April 2009). <a href="https://web.williams.edu/Mathematics/csilva/NonsingularET_Apr.pdf">Ergodic theory: Nonsingular transformations</a>; See <a href="https://arxiv.org/abs/0803.2424">Arxiv arXiv:0803.2424</a>.</li>
<li>Krengel, Ulrich (1985), Ergodic theorems, De Gruyter Studies in Mathematics, vol. 6, de Gruyter, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-11-008478-3</li></ul>

Wandering set open-in-new

Wandering set