Fixed-point index

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the fixed-point index is a concept in <a href="/facts/Topological/2EqW47cU">topological</a> <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed-point</a> theory, and in particular <a href="/facts/Nielsen_theory/1VvcXSSb">Nielsen theory</a>. The fixed-point index can be thought of as a <a href="/facts/Multiplicity_(mathematics)/dndUyetA">multiplicity</a> measurement for fixed points.
The index can be easily defined in the setting of <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a>: Let f(z) be a <a href="/facts/Holomorphic_map/kVzthtEf">holomorphic mapping</a> on the complex plane, and let z0 be a fixed point of f. Then the function f(z) − z is holomorphic, and has an isolated zero at z0. We define the fixed-point index of f at z0, denoted i(f, z0), to be the multiplicity of the zero of the function f(z) − z at the point z0.
In real Euclidean space, the fixed-point index is defined as follows: If x0 is an isolated fixed point of f, then let g be the function defined by

g
        (
        x
        )
        =
        
          
            
              x
              −
              f
              (
              x
              )
            
            
              
                |
              
              
                |
              
              x
              −
              f
              (
              x
              )
              
                |
              
              
                |
              
            
          
        
        .
      
    
    {\displaystyle g(x)={\frac {x-f(x)}{||x-f(x)||}}.}

Then g has an isolated singularity at x0, and maps the boundary of some deleted neighborhood of x0 to the unit sphere. We define i(f, x0) to be the <a href="/facts/L._E._J._Brouwer/6q6ULf3b">Brouwer</a> <a href="/facts/Degree_of_a_continuous_mapping/yWxq9e4B">degree</a> of the mapping induced by g on some suitably chosen small sphere around x0.

Fixed-point index open-in-new

Fixed-point index