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4.1: The fixed points and periodic orbits

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    The first derivative of quadratic map f(z) = z2 + c at a fixed point z∗ = f(z∗ )

        λ = f '(z∗ ) = 2z∗

    is a complex number called multiplier (or the eigenvalue) of the point. In particular, f ' = 0 for critical points of f. For small enough ε

        f(z∗ + ε) = f(z∗ ) + λ ε + O(ε2),

    so a fixed point is either attracting or repelling or indifferent (neutral) according as its multiplier satisfies |λ| < 1 or |λ| > 1 or |λ| = 1.
    It is evident that for a period-n orbit (or cycle)

        f:  z1 → z2 → ... → zn = z1

    n points zi (i = 1,2,...,n) are the fixed points of the n-fold iterate f on(z) = f(f(...f(z))) of map f

        zi = f on(zi ).

    There is product formula for multiplier of a period-n orbit

        λ = (f on)' (zi ) = f '(z1 ) f '(z2 ) ... f '(zn ) = 2 n z1 z2 ... zn .

    Basin of attraction of periodic orbit

    If O is an attracting period-n orbit zi (i = 1,2,...,n), we called the basin of attraction of zi the open set of all points z for which the successive iterates f on(z), f o2n(z),... converge towards zi .

    4.1.1.png
    Figure \(\PageIndex{1}\)
    4.1.2.png
    Figure \(\PageIndex{2}\)

    As we know, for any c in the interior of a (m / n) primary bulb of the Mandelbrot set the corresponding Julia set is connected. It contains infinite sequence of n joint in one point pieces (n = 2 and 3 for the (1/2) and (1/3) bulbs in these pictures). Interior of any of such piece is attracted under iterations of f on(z) to one of points of period-n orbits. Here all pieces that are attracted to the point z1 in the center of the J-set are painted in "red" color. The other pieces (which are attracted to the second and third points of orbit) are painted in "blue" and "green". You see that in any point of the J-set only all n colors can meet together. And we have one order of colors ("red → blue → green" under revolutions in the counter-clockwise direction).

    One more "daisy" below.

    4.1.3.png
    Figure \(\PageIndex{3}\)

     

     

     

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