Number of connected components for the filled julia set of $z^2 + c z^5$

Question

For any polynomial map $f$ we can define the filled Julia $K$ to be closure of the complement of $ \Omega$ in $\mathbb{C}$ of the basin of infinity
$$\Omega = \{z \in \mathbb{C}; f^{\circ n}(z)\rightarrow \infty \}.$$

Thus the boundary of $K$ is what is often called the Julia set. Now there are examples like $f=z^2$ for which the filled Julia set is connected and locally connected.

Lets consider iterations of the form

$$f(z) = z^2 + c \times z^5$$

for complex $z$ and a given positive real $c>0$

For what values $c$ is the Filled Julia set connected with a single component ??

I know that it is connected with a single component for $c=0.23$ but has infinitely many connected components for $c=0.22$

Where is the crossover value of $c$ to go from a single component to infinitely many ?
Does that value have a closed form ?

Do I need to solve a quintic or compute an infinite sum ?

I know -or at least more or less know- that if one finite critical point of a finite number of iterations escapes to infinity, then the filled-in Julia set consists of infinitely many components.

But Im not able to apply it.

I guess I only need to study the zero's of $f'(z)$ to see if they go to infinity ?
But how can I be certain if they will or will not ?

And especially the boundary cross-over value for $c$ ITSELF ; does it have one component or infinitely many ?

---

edit

Using the comment of Claude, I will use his computation.
I will try to make a little logical conclusion.

For critical points,

$$f′(z)=2z+5cz^4=z(2+5cz^3)=0$$

has solutions

$$z=0$$
,
$$z= \sqrt[3]{\frac{−2}{5c}}$$
(1 real, 2 complex (1 conjugate pair)).

Now lets iterate once with this real number.
( 0 is trivial so we try the other real value )

$$z = \sqrt[3]{\frac{−2}{5c}}$$

$$z^2 = \sqrt[3]{\frac{4}{25c^2}}$$

$$z^5 = \sqrt[3]{\frac{−32}{3125c^2}}$$

$$z^2 + c z^5$$
then gives us

$$\sqrt[3]{\frac{4}{25c^2}}+\sqrt[3]{\frac{−32}{3125c^2}}$$
so we solve that by setting it equal to $0$ or the starting value $z$ above :

$$\sqrt[3]{\frac{4}{25c^2}}+\sqrt[3]{\frac{−32}{3125c^2}}=0$$

no real solutions.

Further

$$\sqrt[3]{\frac{4}{25c^2}}+\sqrt[3]{\frac{−32}{3125c^2}} = \sqrt[3]{\frac{−2}{5c}}$$

This has 1 real solution :

$$c = – \frac{54}{625}$$

although this is actually negative and I asked about real $c>0$, we can conclude

$$f(z) = z^2 – \frac{54}{625} \times z^5$$

has a filled Julia set with exactly one connected component.

However this is also true of

$$f(z) = z^2 – \frac{54}{624} \times z^5$$

and

$$f(z) = z^2 – \frac{54}{626} \times z^5$$

So nothing special.

The question remains open.

Maybe we should use derivatives again or maybe we should use geometric function theory.

And maybe we should use a higher number of iterations before solving equations.
But that might get more complicated.

I have not considered the non-real solutions $c$ or $z_0$ since the question was about real $c$ but maybe it will help ? I dont know.

---

edit2

$$f(z) = z^2 – 0.07 \times z^5$$

has one component.

and

$$f(z) = z^2 – 0.03 \times z^5$$

has infinitely many components.

so $c$ around $0.2$ is not the only real interesting value.

Again, I have no closed form for this boundary value of $c$ between $-0.03$ and $-0.07$.

Nor do I know if the boundary values around $-0.07$ and $0.2$ are related by an algebraic equation.

It seems there are 5 intervals for $c$ more or less like this

$[-oo,-0.05[$ has one component.

$[-0.05,0[$ has infinitely many components.

$0$ has one component.

$]0,0.225[$ has infinitely many components.

$[0.225,+oo]$ has one component.

Which probably makes sense for a degree $5$ iteration.

---

Answer 1

Conclusion

The special point is at $$c = 0.2294487977025897\ldots$$ satisfying $$f_c(f_c(z_0(c))) = f_c(z_0(c))$$ where $$z_0(c) = \sqrt[3]{-\frac{2}{5c}}$$ is the non-zero real critical point. Its filled-in Julia set has a single component.

Finding the point

Script for (wx)Maxima computer algebra system:

f(c, z) := z^2 + c * z^5;
critical_points : solve(diff(f(c, z), z) = 0, z);
front(l) := firstn(l, length(l) - 1);
solutions : unique(xreduce(append, front(map
  ( lambda([cp], solve([at(f(c, f(c, z)) = f(c, z), cp)], [c]))
  , critical_points
  ))));
find_root(solutions[1], c, 0.2, 0.3);

Being more explicit:

solve([at(f(c, f(c, z)) = f(c, z), critical_points[3])], [c]);
%[2];
% * 9375 * 2^(1/3);
% + 81*2^(4/3);
%^2;
expand(%);
at(%, [c = x^3]);
solve(%, x);
%[2];
at(%, [x = c^(1/3)]);

gives the equation $$5^{35/3}c^{5/3}-5^{10}\cdot3\cdot2^{2/3}c-5^5\cdot3^4\cdot2^{8/3} = 0$$

whose real root $c \in [0.2, 0.3]$ seems to match the solution found in the previous code block.

So, yes, you need to solve a quintic (a polynomial in $x = c^{1/3}$).

Finding the number of components

Consider the long-term behaviour of (one of) the other non-zero complex conjugate pair of critical points to know how many components the Julia set has for this specific $c$. Using SageMath I found that the other non-zero critical point $z_1(c)$ is pre-periodic, with $$f_c(z_1(c)) = f_c(f_c(f_c(z_1(c))))$$ so it remains bounded. As no critical points escape to infinity, the Julia set for this $c$ value has a single component.

Script for SageMath:

x = polygen(QQbar)
f(c, z) = z^2 + c * z^5
five3 = QQbar.polynomial_root(x^3 - 5, CIF(RIF(0, 2), RIF(-1, 1)))
two3 = QQbar.polynomial_root(x^3 - 2, CIF(RIF(0, 2), RIF(-1, 1)))
P = five3^35 * x^5 - (5^10 * 3 * two3^2) * x^3 - 5^5 * 3^4 * two3^8
c0 = QQbar.polynomial_root(P, CIF(RIF(0.6, 0.7), RIF(-0.1, 0.1)))^3
print(c0)
z0 = QQbar(- (2 / (5 * c0))^(1/3))
z1 = QQbar((- 2 / (5 * c0))^(1/3))
print(z0, f(c0, z0), f(c0, f(c0, z0)))
print(z1, f(c0, z1), f(c0, f(c0, z1)), f(c0, f(c0, f(c0, z1))))
print(all([f(c0, z0) == f(c0, f(c0, z0)), f(c0, z1) == f(c0, f(c0, f(c0, z1)))]))

The point is in the right place

From the left

Arbitrarily close nearby values of $c$ less than the special point give filled-in Julia sets with infinitely many components.

Considering small real perturbations in $c$ near the special point and non-zero real critical point, the difference in $z$ has the opposite sign to the difference in $c$. Now considering small real perturbations in $z$ near the eventually fixed point in the limit of the perturbation in $c$ going to $0$, if the perturbation in $z$ is initially positive, then it grows without bound, eventually escaping towards infinity.

Continuing with SageMath:

zs = f(c0, z0)
F(C, Z, c, z) = (f(C + c, Z + z) - f(C, Z)).expand()
print(F(c0, z0, c, 0)) # F has opposite sign to c
print(F(c0, zs, 0, z)) # F > 2z if z > 0

From the right

It remains to be shown that arbitrarily close nearby real values of $c$ greater than the special point give filled-in Julia sets with a single component.

If we have $|z_{n+1}| < |z_n| < \epsilon$, then iterations are certainly bounded as $n \to \infty$. Substituting, we get $$|\epsilon^2 + c \epsilon^5| < \epsilon^2 + |c| \epsilon^5 < \epsilon $$

And if $|c| > \frac{2}{5\epsilon^3}$ then $z_0 < \epsilon$. So $$\frac{2}{5\epsilon^3} < |c| < \frac{1 - \epsilon^3}{\epsilon^4}$$ whence $$5 \epsilon^3 + 2 \epsilon < 5,$$ that is $$\epsilon < \epsilon_{\text{max}} \approx 0.86755924139011\ldots$$ and $$|c| > R_{\text{min}} \approx 0.6125796570109\ldots$$

That still leaves a big gap of positive real $c$ that haven't been proven to remain bounded.

Visualisation

Here's the "Mandelbrot set" plot of the $c$ plane with the proven-bounded part in red, points that definitely escaped to infinity in white (these have filled-in Julia sets with infinitely many components, the rest have a single component), and the remainder in black:

Here's a zoom-in of the previous image with a grid overlaid in green (spacing 0.1 units, the axes are slightly thicker); the special $c$ is at the right-most corner of the white region:

Here's an animation of the filled in Julia set (black) for the special $c \pm \frac{1}{10}$.

Parameter plane GLSL fragment shader source code snippet for FragM (fork of Fragmentarium):

#version 330 core

#include "TwoD.frag"

const float pi = 3.141592653;

vec2 cMul(vec2 a, vec2 b)
{
  return vec2(a.x * b.x - a.y * b.y, a.x * b.y + a.y * b.x);
}

vec2 cConj(vec2 a)
{
  return vec2(a.x, -a.y);
}

vec2 cDiv(vec2 a, vec2 b)
{
  return cMul(a, cConj(b) / dot(b, b));
}

vec2 cExp(vec2 a)
{
  return exp(a.x) * vec2(cos(a.y), sin(a.y));
}

vec2 cLog(vec2 a)
{
  return vec2(log(length(a)), atan(a.y, a.x));
}

vec2 cCbrt(vec2 a, int n)
{
  return cExp((cLog(a) + vec2(0, float(n) * 2.0 * pi)) / 3.0);
}

vec3 color(vec2 c, vec2 dx, vec2 dy) {
  bvec3 escaped = bvec3(false);
  for (int j = 0; j < 3; ++j)
  {
    vec2 z = cCbrt(cDiv(vec2(-2.0/5.0, 0.0), c), j);
    for (int i = 0; i < 1000; ++i)
    {
      escaped[j] = escaped[j] || !(dot(z, z) < 1.0e6);
      if (escaped[j]) break;
      vec2 z2 = cMul(z, z);
      vec2 z4 = cMul(z2, z2);
      vec2 cz4 = cMul(c, z4);
      z = cMul(z, z + cz4);
    }
  }
  vec2 g = c;
  if (abs(g.y) < dy.y) return vec3(0.0, 1.0, 0.0);
  if (abs(g.x) < dy.y) return vec3(0.0, 1.0, 0.0);
  g *= 10.0;
  g += 0.5;
  g -= floor(g);
  g -= 0.5;
  if (abs(g.y) < 2 * dy.y) return vec3(0.0, 1.0, 0.0);
  if (abs(g.x) < 2 * dy.y) return vec3(0.0, 1.0, 0.0);
  return mix(vec3(escaped), vec3(1.0, 0.0, 0.0), float((length(c) > 0.61257965701099)));
}

Filled-in Julia set GLSL fragment shader source code snippet for FragM:

#version 330 core

#include "TwoD.frag"

const float pi = 3.141592653;

vec2 cMul(vec2 a, vec2 b)
{
  return vec2(a.x * b.x - a.y * b.y, a.x * b.y + a.y * b.x);
}

vec3 color(vec2 z, vec2 dx, vec2 dy) {
  vec2 c = vec2(0.2294487977025897 - 0.1 * cos(2.0 * pi * time), 0.0);
  bool escaped = false;
  for (int i = 0; i < 1000; ++i)
  {
    escaped = escaped || !(dot(z, z) < 1.0e6);
    if (escaped) break;
    vec2 z2 = cMul(z, z);
    vec2 z4 = cMul(z2, z2);
    vec2 cz4 = cMul(c, z4);
    z = cMul(z, z + cz4);
  }
  return vec3(escaped);
}

Reproducibility

Try the SageMath scripts online here. Today the output is:

0.2294487977025897? + 0.?e-37*I
-1.203533181541392? 0.8690952714426872? 0.8690952714426872?
0.6017665907706960? + 1.042290309512355?*I -0.4345476357213436? + 0.752658583378300?*I -0.4345476357213436? - 0.752658583378300?*I -0.4345476357213436? + 0.752658583378300?*I
True
-2.525168374948163?*c
0.2294487977025897?*z^5 + 0.9970643256076521?*z^4 + 1.733087781419605?*z^3 + 2.506218395826876?*z^2 + 2.392714185671939?*z

I used the Debian package for maxima, version 5.46.0-8 (not the latest version available in the Bookworm repositories).

I used https://sagecell.sagemath.org as interface to SageMath.

Images were rendered with Fragmentarium 2.5.7.221026 Digilantism with my example files (include path: Examples/Include;Examples/Claude).