z^4+2cz^2-z+c^2+c=0.Since a fixed point for Q_c are also roots of this polynomial, we can factorize via long division and obtain
z^4+2cz^2-z+c^2+c=(z^2-z+c)(z^2+z+c+1)=0.Hence, Q_c must have periodic points of prime period 2 at each root of the polynomial z^2+z+c+1=0.
z+=1/2(-1 +sqrt[1-4(c+1)]) and z-=1/2(-1-sqrt[1-4(c+1)]).Moreover
| DQ^2_c(z±) | = | DQ_c(z+) |*| DQ_c(z-) | = 2|z+|*2|z-| = 4|z+*z-|since z+ is the conjugate of z-, we get
| DQ^2_c(z±) | = 4 | z+|^2 = 4|1+c|.Then, the set of c-values for which Q_c has an attracting 2-cycle is given by the inequality |1+c| < 1/4.
H°Q_c(z)=z^2+c, and Q_c°H(z)=(z)^2 +c.
The Logistic Functions. The following exercises deal with the logistic family F_a(z)=az(1-z) where both a and z are complex numbers.
|z|*|z-1| > |z|/|a|, that is |az|*|1-z| > |z|.Hence, |F_a(z)| > |z| if |z| > 1/|a| + 1, a=/=0. Notice that 1/|a| + 1 approaches 1 as |a|-> infinity. Hence our criteria works for
|z| > max { 1, 1/|a| + 1},i.e. at least outside the unit circle.
DF_a(z)=0 <==> a-2az=0, <==> z=1/2for all a-values. Since the c.pt. lies inside the unit disk, we cannot apply the coterie given in ex #4.. So we must require that at least, F_z(1/2) lie outside the unit disk. That is,
|F_a(1/2)| > 1/|a| + 1 implies |F^2_a(1/2)| > |F_a(1/2)|and so the orbit of c.pt. will escape to infinity. Thus
|F_a(1/2)| =|a|/2=1/|a| + 1 <==> |a^2|-4|a|-4=0.Computing the roots of this quadratic polynomial, we obtain
a+=2+2sqrt[2] and a-=2-2sqrt[2]< 0,so we discard the last solution. Then we must have |a| > 2+2sqrt[2] to ensure that |F_a(1/2)| > 1/|a| + 1, so it implies
|F^2_a(1/2)| > |F_a(1/2)|.Hence the orbit of 1/2 must escape under F_a.
H°Q_c(z) = u(z^2+V(a))+v = uz^2 + uV(a) + v, andComparing terms, we obtain the next equations
F_a°H(z) = a(uz+v)(1-uz-v) = -auz^2 + auz(2v-1)+v(v-1).
quadratic terms | -au^2=u | equation (1) |
linear terms | -au(2v-1) = 0 | equation (2) |
constant terms | -v(v-1)=v+u(a/2+a-^2/4) | equation (3) |
From (1), u=0 or u=-1/a. But if u=0, H(z) reduces to a constant function!, hence we must only consider u=-1/a.
From (2), we obtain v=1/2 (now, equation (3) determines two more values for v, compute them and define the maps H(z) associated to those values. Are those maps also conjugacies?).
Hence H(z)=-z/a + 1/2 and is clearly a homeomorphism. Checking the conjugacy equation, we get
H°Q_c(z) = -1/a (z^2 +a/2-a^2/4)+1/2 = -z^2/a +a/4, andNow, if a=0, the correspondence V(a) send zero into zero. But while Q_0 reduces to z^2, F_0 reduces to the constant zero map, hence with no dynamics to conjugate with z^2.
F_a°H(z) = a(-z/a+1/2)(a+z?a-1/2) = -z^2/a + a/4.
z_1=0 and z_2=(a-1)/a.To determine when we have an attracting fixed point, we need to compute | DF_a(z) | = |a|*|1-2z| for z_1 and z_2.
For z_1, we obtain |DF_a(0)| = |a|. Hence, the origin is attracting <==> |a|<1. Hence L_1 is the unit disk centered at the origin.
For z_2, we obtain |DF_a(z_2)| = |2-a|. Hence, the second fixed point
is attracting for any a-value that lies inside the disk centered at 2 and
with radius 1. This disk is L_1 for the second fixed point.
V(exp(it)) = exp(it)/2 - exp(2it)/4which is the equation of the main cardioid as given in page 252. Hence V takes L_1(0) into the main cardioid of the Mandelbrot set.
Now, parametrize the boundary of L_1 for the second fixed point by exp(it)-2, again 0 <= t <= 2pi. Then
V(exp(it)-2) = exp(it)/2 +1 - exp(2it)/ 4 - exp(it) -1 = - exp(it)/2 -exp(2it)/4That is, V takes L_1(z_2) into the main cardioid of the Mandelbrot set, but runs through the curve with the clockwise direction.
c = a/2 - a^2/4for a in terms of c. Then we obtain a quadratic a^2 -2a + 4c = 0 and two roots that are the inverse maps for V
U+(c) = 1 + sqrt[1-4c] and U-(c) = 1- sqrt[1-4c].Hence, the preimages under V of the 2-bulb in the Mandelbrot set are given by two circles in the a-plane, with centers at U+(-1)=1+sqrt[5] and U-(-1)=1-sqrt[5].
Higher Degree Polynomials. The following eight exercises deal with polynomials of the form P_(d,c)(z)=z^d+c.
DP_(d,c)(z) = 0 <==> dz^(d-1) = 0 <==> d=0 or z=0.Hence, for d>1, DP_(d,c) is a polynomial of degree d-1, and z=0 is the unique critical point of P_(d,c).
Assume |z| >= c and |z|^(d-1) > 2. We want to show
|P_(d,c)(z)| > |z|.To do so, notice that since |z|^(d-1) > 2, then |z|^d > 2|z| (provided z=/=0). Then
|z|^d - |z| > |z| ==> |z|^d - |c| > = |z|^d - |z| > |z|.By the triangle inequality,
|z| < |z|^d - |c| = < |z^d+c| = | P_(d,c)(z) | .Hence, if |z| >= |c| and |z|^(d-1) > 2, then P^n_(d,c) diverges as n --> infinity.
|c|^(d-1) > 2,which is the hypothesis given after taking the (d-1)-root. Hence
|P^2_(d,c)(0)| > |P_(d,c)(0)| = |c|,and so the orbit of zero escapes.