Ruelle zeta function

<h2 id="formal-definition">Formal definition</h2>
Let f be a function defined on a <a href="/facts/Manifold/rXPERJtu">manifold</a> M, such that the set of <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed points</a> Fix(f n) is finite for all n > 1. Further let φ be a function on M with values in d × d complex matrices. The zeta function of the first kind is<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

ζ
        (
        z
        )
        =
        exp
        ⁡
        
          (
          
            
              ∑
              
                m
                ≥
                1
              
            
            
              
                
                  z
                  
                    m
                  
                
                m
              
            
            
              ∑
              
                x
                ∈
                Fix
                ⁡
                (
                
                  f
                  
                    m
                  
                
                )
              
            
            Tr
            ⁡
            
              (
              
                
                  ∏
                  
                    k
                    =
                    0
                  
                  
                    m
                    −
                    1
                  
                
                φ
                (
                
                  f
                  
                    k
                  
                
                (
                x
                )
                )
              
              )
            
          
          )
        
      
    
    {\displaystyle \zeta (z)=\exp \left(\sum _{m\geq 1}{\frac {z^{m}}{m}}\sum _{x\in \operatorname {Fix} (f^{m})}\operatorname {Tr} \left(\prod _{k=0}^{m-1}\varphi (f^{k}(x))\right)\right)}

<h2 id="examples">Examples</h2>
In the special case d = 1, φ = 1, we have<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

ζ
        (
        z
        )
        =
        exp
        ⁡
        
          (
          
            
              ∑
              
                m
                ≥
                1
              
            
            
              
                
                  z
                  
                    m
                  
                
                m
              
            
            
              |
              
                Fix
                ⁡
                (
                
                  f
                  
                    m
                  
                
                )
              
              |
            
          
          )
        
      
    
    {\displaystyle \zeta (z)=\exp \left(\sum _{m\geq 1}{\frac {z^{m}}{m}}\left|\operatorname {Fix} (f^{m})\right|\right)}

which is the <a href="/facts/Artin%25E2%2580%2593Mazur_zeta_function/vaR7IcvR">Artin–Mazur zeta function</a>.
The <a href="/facts/Ihara_zeta_function/1FJ8MMr9">Ihara zeta function</a> is an example of a Ruelle zeta function.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/List_of_zeta_functions/lf43angF">List of zeta functions</a></li></ul>

<ul><li>Lapidus, Michel L.; van Frankenhuijsen, Machiel (2006). Fractal geometry, complex dimensions and zeta functions. Geometry and spectra of fractal strings. Springer Monographs in Mathematics. New York, NY: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-33285-5. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1119.28005">1119.28005</a>.</li>
<li><a href="/facts/Motoko_Kotani/6RfYmeeL">Kotani, Motoko</a>; <a href="/facts/Toshikazu_Sunada/8lkWp16o">Sunada, Toshikazu</a> (2000). "Zeta functions of finite graphs". J. Math. Sci. Univ. Tokyo. 7: 7–25.</li>
<li><a href="/facts/Audrey_Terras/pmVYDCq9">Terras, Audrey</a> (2010). Zeta Functions of Graphs: A Stroll through the Garden. Cambridge Studies in Advanced Mathematics. Vol. 128. <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-11367-0. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1206.05003">1206.05003</a>.</li>
<li><a href="/facts/David_Ruelle/mjoK3TGl">Ruelle, David</a> (2002). <a href="https://www.ams.org/notices/200208/fea-ruelle.pdf">"Dynamical Zeta Functions and Transfer Operators"</a> (PDF). Bulletin of AMS. 8 (59): 887–895.</li></ul>

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

<ol>
<li id="fn:1">Terras (2010) p. 28 <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Terras (2010) p. 28 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Terras (2010) p. 29 <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
</ol>

Ruelle zeta function open-in-new

Ruelle zeta function