Lefschetz zeta function

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the <a href="/facts/Lefschetz/Ut2VFwGr">Lefschetz</a> <a href="/facts/Zeta_function/lf43angF">zeta-function</a> is a tool used in topological periodic and <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed point</a> theory, and <a href="/facts/Dynamical_systems/782gREl5">dynamical systems</a>. Given a <a href="/facts/Continuous_map/sKbl02pB">continuous map</a> 
  
    
      
        f
        :
        X
        →
        X
      
    
    {\displaystyle f\colon X\to X}
  
, the zeta-function is defined as the formal series
</p>

ζ
          
            f
          
        
        (
        t
        )
        =
        exp
        ⁡
        
          (
          
            
              ∑
              
                n
                =
                1
              
              
                ∞
              
            
            L
            (
            
              f
              
                n
              
            
            )
            
              
                
                  t
                  
                    n
                  
                
                n
              
            
          
          )
        
        ,
      
    
    {\displaystyle \zeta _{f}(t)=\exp \left(\sum _{n=1}^{\infty }L(f^{n}){\frac {t^{n}}{n}}\right),}

<p>where 
  
    
      
        L
        (
        
          f
          
            n
          
        
        )
      
    
    {\displaystyle L(f^{n})}
  
 is the <a href="/facts/Lefschetz_number/TEjN2DQ8">Lefschetz number</a> of the 
  
    
      
        n
      
    
    {\displaystyle n}
  
-th <a href="/facts/Iterated_function/Vzsx64An">iterate</a> of 
  
    
      
        f
      
    
    {\displaystyle f}
  
. This zeta-function is of note in topological periodic point theory because it is a single invariant containing information about all iterates of 
  
    
      
        f
      
    
    {\displaystyle f}
  
.
</p>

Lefschetz zeta function open-in-new

Lefschetz zeta function