Classification of Fatou components

<h2 id="rational-case">Rational case</h2>
<p>If f is a <a href="/facts/Rational_function/2nJGu0Ko">rational function</a>
</p>

f
        =
        
          
            
              P
              (
              z
              )
            
            
              Q
              (
              z
              )
            
          
        
      
    
    {\displaystyle f={\frac {P(z)}{Q(z)}}}

<p>defined in the <a href="/facts/Extended_complex_plane/4XgBzplC">extended complex plane</a>, and if it is a nonlinear function (degree > 1)
</p>

d
        (
        f
        )
        =
        max
        (
        deg
        ⁡
        (
        P
        )
        ,
        
        deg
        ⁡
        (
        Q
        )
        )
        ≥
        2
        ,
      
    
    {\displaystyle d(f)=\max(\deg(P),\,\deg(Q))\geq 2,}

<p>then for a periodic <a href="/facts/Connected_component_(analysis)/9qhzYyT7">component</a> 
  
    
      
        U
      
    
    {\displaystyle U}
  
 of the <a href="/facts/Fatou_set/dmn8xIcG">Fatou set</a>, exactly one of the following holds:
</p>
<ol><li>
  
    
      
        U
      
    
    {\displaystyle U}
  
 contains an <a href="/facts/Attracting_periodic_point/9tRHxcrn">attracting periodic point</a></li>
<li>
  
    
      
        U
      
    
    {\displaystyle U}
  
 is parabolic<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a></li>
<li>
  
    
      
        U
      
    
    {\displaystyle U}
  
 is a <a href="/facts/Siegel_disc/xTXJh5RV">Siegel disc</a>:  a simply connected Fatou component on which <i>f</i>(<i>z</i>) is analytically conjugate to a Euclidean rotation of the unit disc onto itself by an irrational rotation angle.</li>
<li>
  
    
      
        U
      
    
    {\displaystyle U}
  
 is a <a href="/facts/Herman_ring/Qx4qAvRs">Herman ring</a>: a double connected Fatou component (an <a href="/facts/Annulus_(mathematics)/rhxKuoem">annulus</a>) on which <i>f</i>(<i>z</i>) is analytically conjugate to a Euclidean rotation of a round annulus, again by an irrational rotation angle.</li></ol>

<h3>Attracting periodic point</h3>
<p>The components of the map 
  
    
      
        f
        (
        z
        )
        =
        z
        −
        (
        
          z
          
            3
          
        
        −
        1
        )
        
          /
        
        3
        
          z
          
            2
          
        
      
    
    {\displaystyle f(z)=z-(z^{3}-1)/3z^{2}}
  
  contain the attracting points that are the solutions to 
  
    
      
        
          z
          
            3
          
        
        =
        1
      
    
    {\displaystyle z^{3}=1}
  
. This is because the map is the one to use for finding solutions to the equation 
  
    
      
        
          z
          
            3
          
        
        =
        1
      
    
    {\displaystyle z^{3}=1}
  
 by <a href="/facts/Newton%E2%80%93Raphson/VlRhI2FL">Newton–Raphson</a> formula. The solutions must naturally be attracting fixed points.
</p>

<h3>Herman ring</h3>
<p>The map
</p>

f
        (
        z
        )
        =
        
          e
          
            2
            π
            i
            t
          
        
        
          z
          
            2
          
        
        (
        z
        −
        4
        )
        
          /
        
        (
        1
        −
        4
        z
        )
      
    
    {\displaystyle f(z)=e^{2\pi it}z^{2}(z-4)/(1-4z)}

<p>and t = 0.6151732... will produce a Herman ring.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> It is shown by <a href="/facts/Mitsuhiro_Shishikura/n4BolVca">Shishikura</a> that the degree of such map must be at least 3, as in this example.
</p>
<h3>More than one type of component</h3>
<p>If degree d is greater than 2 then there is more than one critical point and then can be more than one type of component
</p>

<h2 id="transcendental-case">Transcendental case</h2>
<h3>Baker domain</h3>
<p>In case of <a href="/facts/Transcendental_functions/yMkeq4Eo">transcendental functions</a> there is another type of periodic Fatou components, called Baker domain: these are "<a href="/facts/Domain_(mathematical_analysis)/xT44oI3E">domains</a> on which the iterates tend to an <a href="/facts/Essential_singularity/KMb1R292">essential singularity</a> (not possible for polynomials and rational functions)"<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> one example of such a function is:<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>

f
        (
        z
        )
        =
        z
        −
        1
        +
        (
        1
        −
        2
        z
        )
        
          e
          
            z
          
        
      
    
    {\displaystyle f(z)=z-1+(1-2z)e^{z}}

</p>
<h3>Wandering domain</h3>
<p>Transcendental maps may have <a href="/facts/Wandering_set/sclWs7QL">wandering domains</a>: these are Fatou components that are not eventually periodic.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/No-wandering-domain_theorem/uuMth8xv">No-wandering-domain theorem</a></li>
<li><a href="/facts/Montel%27s_theorem/JtG0GBXE">Montel's theorem</a></li>
<li><a href="/facts/Fritz_John/45S8VX2o">John</a> Domains<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></li>
<li><a href="/facts/Attractor/FAx5Db4w">Basins of attraction</a></li></ul>

<ul><li><a href="/facts/Lennart_Carleson/drIokpo0">Lennart Carleson</a> and Theodore W. Gamelin, <i>Complex Dynamics</i>, Springer 1993.</li>
<li><a href="/facts/Alan_F._Beardon/7j8kUweA">Alan F. Beardon</a> <i>Iteration of Rational Functions</i>, Springer 1991.</li></ul>

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

<ol>
<li id="fn:1"><p>wikibooks : parabolic Julia sets <a href="https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/parabolic" target="_blank">https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/parabolic</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Milnor, John W. (1990), Dynamics in one complex variable, arXiv:math/9201272, Bibcode:1992math......1272M <a href="/wiki/John_Milnor" target="_blank">/wiki/John_Milnor</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>An Introduction to Holomorphic Dynamics  (with particular focus on transcendental functions)by L. Rempe <a href="http://pcwww.liv.ac.uk/~lrempe/Talks/liverpool_seminar_2006.pdf" target="_blank">http://pcwww.liv.ac.uk/~lrempe/Talks/liverpool_seminar_2006.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Siegel Discs in Complex Dynamics by Tarakanta Nayak <a href="http://www.ncnsd.org/proceedings/proceeding05/paper/185.pdf" target="_blank">http://www.ncnsd.org/proceedings/proceeding05/paper/185.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>A transcendental family with Baker domains by Aimo Hinkkanen, Hartje Kriete and Bernd Krauskopf
 <a href="http://www.math.uiuc.edu/~aimo/anim.html" target="_blank">http://www.math.uiuc.edu/~aimo/anim.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>JULIA AND JOHN REVISITED by NICOLAE MIHALACHE <a href="https://arxiv.org/abs/0803.3889" target="_blank">https://arxiv.org/abs/0803.3889</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
</ol>

Classification of Fatou components open-in-new

Classification of Fatou components