Collage theorem

<h2 id="statement">Statement</h2>
<p>Let 
  
    
      
        
          X
        
      
    
    {\displaystyle \mathbb {X} }
  
 be a <a href="/facts/Complete_metric_space/lsckmU8G">complete metric space</a>. 
Suppose 
  
    
      
        L
      
    
    {\displaystyle L}
  
 is a nonempty, <a href="/facts/Compact_subset/d0cgXJH7">compact subset</a> of 
  
    
      
        
          X
        
      
    
    {\displaystyle \mathbb {X} }
  
 and let 
  
    
      
        ϵ
        >
        0
      
    
    {\displaystyle \epsilon >0}
  
 be given. 
Choose an <a href="/facts/Iterated_function_system/j56g6yEM">iterated function system</a> (IFS) 
  
    
      
        {
        
          X
        
        ;
        
          w
          
            1
          
        
        ,
        
          w
          
            2
          
        
        ,
        …
        ,
        
          w
          
            N
          
        
        }
      
    
    {\displaystyle \{\mathbb {X} ;w_{1},w_{2},\dots ,w_{N}\}}
  
 with contractivity factor  
  
    
      
        s
        ,
      
    
    {\displaystyle s,}
  
 where 
  
    
      
        0
        ≤
        s
        <
        1
      
    
    {\displaystyle 0\leq s<1}
  
 (the contractivity factor 
  
    
      
        s
      
    
    {\displaystyle s}
  
 of the IFS is the maximum of the contractivity factors of the maps 
  
    
      
        
          w
          
            i
          
        
      
    
    {\displaystyle w_{i}}
  
).  Suppose
</p>

h
        
          (
          
            L
            ,
            
              ⋃
              
                n
                =
                1
              
              
                N
              
            
            
              w
              
                n
              
            
            (
            L
            )
          
          )
        
        ≤
        ε
        ,
      
    
    {\displaystyle h\left(L,\bigcup _{n=1}^{N}w_{n}(L)\right)\leq \varepsilon ,}

<p>where 
  
    
      
        h
        (
        ⋅
        ,
        ⋅
        )
      
    
    {\displaystyle h(\cdot ,\cdot )}
  
 is the <a href="/facts/Hausdorff_distance/3tKbFmKu">Hausdorff metric</a>. Then
</p>

h
        (
        L
        ,
        A
        )
        ≤
        
          
            ε
            
              1
              −
              s
            
          
        
      
    
    {\displaystyle h(L,A)\leq {\frac {\varepsilon }{1-s}}}

<p>where <i>A</i> is the attractor of the IFS. Equivalently,
</p>

h
        (
        L
        ,
        A
        )
        ≤
        (
        1
        −
        s
        
          )
          
            −
            1
          
        
        h
        
          (
          
            L
            ,
            
              ∪
              
                n
                =
                1
              
              
                N
              
            
            
              w
              
                n
              
            
            (
            L
            )
          
          )
        
        
      
    
    {\displaystyle h(L,A)\leq (1-s)^{-1}h\left(L,\cup _{n=1}^{N}w_{n}(L)\right)\quad }
  
, for all nonempty, compact subsets L of 
  
    
      
        
          X
        
      
    
    {\displaystyle \mathbb {X} }
  
.
<p>Informally, If 
  
    
      
        L
      
    
    {\displaystyle L}
  
 is close to being stabilized by the IFS, then 
  
    
      
        L
      
    
    {\displaystyle L}
  
 is also close to being the attractor of the IFS.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Michael_Barnsley/I5A97LMW">Michael Barnsley</a></li>
<li><a href="/facts/Barnsley_fern/6AJlsLD3">Barnsley fern</a></li></ul>

<ul><li>Barnsley, Michael. (1988). <a href="https://archive.org/details/fractalseverywhe0000barn"><i>Fractals Everywhere</i></a>. Academic Press, Inc. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-079062-9.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://www.cut-the-knot.org/ctk/ifs.shtml">A description of the collage theorem and interactive Java applet</a> at <a href="/facts/Cut-the-knot/WLgV43jd">cut-the-knot</a>.</li>
<li><a href="https://web.archive.org/web/20080917024731/http://www.math.okstate.edu/mathdept/dynamics/lecnotes/node47.html">Notes on designing IFSs to approximate real images.</a></li>
<li><a href="https://web.archive.org/web/20100606074948/http://scimath.unl.edu/MIM/files/MATExamFiles/Snyder_MAT_Exam_ExpositoryPaper.pdf">Expository Paper on Fractals and Collage theorem</a></li></ul>

Collage theorem open-in-new

Collage theorem