Kolmogorov automorphism

<h2 id="formal-definition">Formal definition</h2>
<p>Let 
  
    
      
        (
        X
        ,
        
          
            B
          
        
        ,
        μ
        )
      
    
    {\displaystyle (X,{\mathcal {B}},\mu )}
  
 be a <a href="/facts/Standard_probability_space/rtVwnswY">standard probability space</a>, and let 
  
    
      
        T
      
    
    {\displaystyle T}
  
 be an invertible, <a href="/facts/Measure-preserving_transformation/acIdlpqt">measure-preserving transformation</a>.  Then 
  
    
      
        T
      
    
    {\displaystyle T}
  
 is called a <i>K</i>-automorphism, <i>K</i>-transform or <i>K</i>-shift, if there exists a sub-<a href="/facts/Sigma_algebra/8LJINLNc">sigma algebra</a> 
  
    
      
        
          
            K
          
        
        ⊂
        
          
            B
          
        
      
    
    {\displaystyle {\mathcal {K}}\subset {\mathcal {B}}}
  
 such that the following three properties hold:
</p>

(1) 
          
        
        
          
            K
          
        
        ⊂
        T
        
          
            K
          
        
      
    
    {\displaystyle {\mbox{(1) }}{\mathcal {K}}\subset T{\mathcal {K}}}

(2) 
          
        
        
          ⋁
          
            n
            =
            0
          
          
            ∞
          
        
        
          T
          
            n
          
        
        
          
            K
          
        
        =
        
          
            B
          
        
      
    
    {\displaystyle {\mbox{(2) }}\bigvee _{n=0}^{\infty }T^{n}{\mathcal {K}}={\mathcal {B}}}

(3) 
          
        
        
          ⋂
          
            n
            =
            0
          
          
            ∞
          
        
        
          T
          
            −
            n
          
        
        
          
            K
          
        
        =
        {
        X
        ,
        ∅
        }
      
    
    {\displaystyle {\mbox{(3) }}\bigcap _{n=0}^{\infty }T^{-n}{\mathcal {K}}=\{X,\varnothing \}}

<p>Here, the symbol 
  
    
      
        ∨
      
    
    {\displaystyle \vee }
  
 is the <a href="/facts/Join_(sigma_algebra)/8LJINLNc">join of sigma algebras</a>, while 
  
    
      
        ∩
      
    
    {\displaystyle \cap }
  
 is <a href="/facts/Set_intersection/RPfUNJh9">set intersection</a>. The equality should be understood as holding <a href="/facts/Almost_everywhere/1quRFtJE">almost everywhere</a>, that is, differing at most on a set of <a href="/facts/Measure_zero/kMudL8Mn">measure zero</a>.
</p>
<h2 id="properties">Properties</h2>
<p>Assuming that the sigma algebra is not trivial, that is, if 
  
    
      
        
          
            B
          
        
        ≠
        {
        X
        ,
        ∅
        }
      
    
    {\displaystyle {\mathcal {B}}\neq \{X,\varnothing \}}
  
, then 
  
    
      
        
          
            K
          
        
        ≠
        T
        
          
            K
          
        
        .
      
    
    {\displaystyle {\mathcal {K}}\neq T{\mathcal {K}}.}
  
  It follows that <i>K</i>-automorphisms are <a href="/facts/Strong_mixing/ttsAJCID">strong mixing</a>.
</p><p>All <a href="/facts/Bernoulli_automorphism/JA8N69xf">Bernoulli automorphisms</a> are <i>K</i>-automorphisms, but not <i>vice versa</i>.
</p><p>Kolmogorov automorphisms are precisely the natural extensions of exact endomorphisms,<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> i.e. mappings 
  
    
      
        T
      
    
    {\displaystyle T}
  
 for which 
  
    
      
        
          ⋂
          
            n
            =
            0
          
          
            ∞
          
        
        
          T
          
            −
            n
          
        
        
          
            M
          
        
      
    
    {\displaystyle \bigcap _{n=0}^{\infty }T^{-n}{\mathcal {M}}}
  
 consists of measure-zero sets or their complements, where 
  
    
      
        
          
            M
          
        
      
    
    {\displaystyle {\mathcal {M}}}
  
 is the sigma-algebra of measureable sets,.
</p>

<h2 id="further-reading">Further reading</h2>
<ul><li>Christopher Hoffman, "<a href="https://www.ams.org/journals/tran/1999-351-10/S0002-9947-99-02446-0/">A K counterexample machine</a>", <i>Trans. Amer. Math. Soc.</i> 351 (1999), pp 4263–4280.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Peter Walters, An Introduction to Ergodic Theory, (1982) Springer-Verlag ISBN 0-387-90599-5 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>V. A. Rohlin, Exact endomorphisms of Lebesgue spaces, Amer. Math. Soc. Transl., Series 2, 39 (1964), 1-36. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Kolmogorov automorphism open-in-new

Kolmogorov automorphism