This article describes periodic points of some complex quadratic maps. 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.
Definitions Let
f c ( z ) = z 2 + c {\displaystyle f_{c}(z)=z^{2}+c\,}
be the complex quadratic mapping, where z {\displaystyle z} and c {\displaystyle c} are complex numbers. Notationally, f c ( k ) ( z ) {\displaystyle f_{c}^{(k)}(z)} is the k {\displaystyle k} -fold composition of f c {\displaystyle f_{c}} with itself (not to be confused with the k {\displaystyle k} th derivative of f c {\displaystyle f_{c}} )—that is, the value after the k-th iteration of the function f c . {\displaystyle f_{c}.} Thus
f c ( k ) ( z ) = f c ( f c ( k − 1 ) ( z ) ) . {\displaystyle f_{c}^{(k)}(z)=f_{c}(f_{c}^{(k-1)}(z)).}
Periodic points of a complex quadratic mapping of period p {\displaystyle p} are points z {\displaystyle z} of the dynamical plane such that
f c ( p ) ( z ) = z , {\displaystyle f_{c}^{(p)}(z)=z,}
where p {\displaystyle p} is the smallest positive integer for which the equation holds at that z. We can introduce a new function:
F p ( z , f ) = f c ( p ) ( z ) − z , {\displaystyle F_{p}(z,f)=f_{c}^{(p)}(z)-z,}
so periodic points are zeros of function F p ( z , f ) {\displaystyle F_{p}(z,f)} : points z satisfying
F p ( z , f ) = 0 , {\displaystyle F_{p}(z,f)=0,}
which is a polynomial of degree 2 p . {\displaystyle 2^{p}.}
Number of periodic points The degree of the polynomial F p ( z , f ) {\displaystyle F_{p}(z,f)} describing periodic points is d = 2 p {\displaystyle d=2^{p}} so it has exactly d = 2 p {\displaystyle d=2^{p}} complex roots (= periodic points), counted with multiplicity.
Stability of periodic points (orbit) - multiplier
The multiplier (or eigenvalue, derivative) m ( f p , z 0 ) = λ {\displaystyle m(f^{p},z_{0})=\lambda } of a rational map f {\displaystyle f} iterated p {\displaystyle p} times at cyclic point z 0 {\displaystyle z_{0}} is defined as:
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