page_title: Periodic points of complex quadratic mappings
section_title: Finite fixed points
hierarchical_section_title: Periodic points of complex quadratic mappings / Period-1 points (fixed points) / Finite fixed points
caption_attribution_description: English: Julia set and external rays landing on fixed point &#x03B1; c {\displaystyle \alpha _{c}\,} . Parametr c is in the center of period 3 hyperbolic component of Mandelbrot set Polski: Zbiór Julia i zewnętrzne promienie lądujące na punkcie stałym &#x03B1; c {\displaystyle \alpha _{c}\,} . Parametr c jest w punkcie centralnym składowej zbioru Mandelbrota o okresie 3.
context_page_description: A map is a formula for computing a value of a variable based on its own previous value or values; a quadratic map is one that involves the previous value raised to the powers one and two; and a complex map is one in which the variable and the parameters are complex numbers. A periodic point of a map is a value of the variable that occurs repeatedly after intervals of a fixed length.
  These periodic points play a role in the theories of Fatou and Julia sets.
context_section_description: Let us begin by finding all finite points left unchanged by one application of . These are the points that satisfy . That is, we wish to solve
  which can be rewritten as
  Since this is an ordinary quadratic equation in one unknown, we can apply the standard quadratic solution formula:
   and 
  So  for    we have two finite fixed points  and .
  Since 
   and  where 
  then .
  Thus fixed points are symmetrical around .
caption: This image shows fixed points (both repelling)
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