We know if R ( z ) is a rational function, then it is in the form R ( z ) = f ( z ) / g ( z ), where f and g are complex polynomials. If we want to find its fixed points, we can take R ( z ) = z, which gives the equation f ( z ) − z ⋅ g ( z ) = 0. By the Fundamental Theorem of Algebra, this equation has finitely many roots; does that mean that those roots are fixed points of R?

Question

We know if   R  (  z  ) is a rational function, then it is in the form   R  (  z  )  =  f  (  z  )      /    g  (  z  ), where   f and   g are complex polynomials. If we want to find its fixed points, we can take   R  (  z  )  =  z, which gives the equation   f  (  z  )  −  z  ⋅  g  (  z  )  =  0. By the Fundamental Theorem of Algebra, this equation has finitely many roots; does that mean that those roots are fixed points of   R?

tennispopj8 · Accepted Answer

You are correct: the fixed points (in       C  ) of the rational fraction   R  (  X  )  =            P      (      X      )              Q      (      X      )       are exactly the roots of the polynomial   P  (  X  )  −  X  Q  (  X  ). In particular, unless   R  (  X  )  =  X, the number of fixed points is finite and bounded by   max  {  deg  ⁡  P  ,  deg  ⁡  Q  +  1  }.

We know if R ( z ) is a rational function, then it is in the form R ( z

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