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 we have . Thus fixed points are symmetrical about . Complex dynamics Here different notation is commonly used: with multiplier and with multiplier Again we have Since the derivative with respect to z is we have This implies that can have at most one attractive fixed point. These points are distinguished by the facts that: is: the landing point of the external ray for angle=0 for the most repelling fixed point of the Julia set the one on the right (whenever fixed point are not symmetrical around the real axis), it is the extreme right point for connected Julia sets (except for cauliflower). is: the landing point of several rays attracting when is in the main cardioid of the Mandelbrot set, in which case it is in the interior of a filled-in Julia set, and therefore belongs to the Fatou set (strictly to the basin of attraction of finite fixed point) parabolic at the root point of the limb of the Mandelbrot set repelling for other values of Special cases An important case of the quadratic mapping is . In this case, we get and . In this case, 0 is a superattractive fixed point, and 1 belongs to the Julia set. Only one fixed point We have exactly when This equation has one solution, in which case . In fact is the largest positive, purely real value for which a finite attractor exists. Infinite fixed point We can extend the complex plane to the Riemann sphere (extended complex plane) by adding infinity: and extend such that Then infinity is: superattracting a fixed point of : Period-2 cycles Period-2 cycles are two distinct points and such that and , and hence for : Equating this to z, we obtain This equation is a polynomial of degree 4, and so has four (possibly non-distinct) solutions. However, we already know two of the solutions. They are and , computed above, since if these points are left unchanged by one application of , then clearly they will be unchanged by more than one application of . Our 4th-order