# Fractals/Iterations in the complex plane/Parameter plane

## Parts of parameter plane

• with respect to the Mandelbrot set
• with respect to the wakes
• inside a p/q wake
• outside any wake (???)

Parts of Mandelbrot set according to M Romera et al.:[1]

• main cardioid
• q/p family (= q/p limb)
• period doubling cascade of hyperbolic components which ends at the Myrberg-Feigenbaum point
• shrub

Not that her q/p not p/q notation is used

# How to choose a point from parameter plane ?

• clicking on parameter points and see what you have ( random choose)
• computing a point with known properties.
• For (parabolic point ) choose hyperbolic component ( period , number) and internal angle (= rotation number) then compute c parameter.
• Misiurewicz points [2]
• morphing
• intresting areas[4]
• zoom

# How to move on parameter plane ?

## Examples

Examples :

 // glsl code by iq from https://www.shadertoy.com/view/Mss3R8
float ltime = 0.5-0.5*cos(time*0.12);
vec2 c = vec2( -0.745, 0.186 ) - 0.045*zoom*(1.0-ltime);


 // glsl code by xylifyx  from https://www.shadertoy.com/view/XssXDr
vec2 c = vec2( 0.37+cos(iTime*1.23462673423)*0.04, sin(iTime*1.43472384234)*0.10+0.50);


 // by Marco Gilardi
vec2 c = vec2(-0.754, 0.05*(abs(cos(0.1*iTime))+0.8));


# Plane types

The phase space of a quadratic map is called its parameter plane. Here:

• ${\displaystyle z0=z_{cr}\,}$ is constant
• ${\displaystyle c\,}$ is variable

There is no dynamics here. It is only a set of parameter values. There are no orbits on the parameter plane.

The parameter plane consists of :

• The Mandelbrot set
• The bifurcation locus = boundary of Mandelbrot set
• Bounded hyperbolic components of the Mandelbrot set = interior of Mandelbrot set [5]

There are many different types of the parameter plane[6] [7]

• plain ( c-plane )
• inverted c-plane = 1/c plane
• lambda plane
• exponential plane ( map) [8][9]
• unrolled plain (flatten' the cardiod = unroll ) [10][11] = "A region along the cardioid is continuously blown up and stretched out, so that the respective segment of the cardioid becomes a line segment. .." ( Figure 4.22 on pages 204-205 of The Science Of Fractal Images)[12]
• transformations [13]
• log : "To illustrate the complexity of the boundary of the Mandelbrot set, Figure 8 renders the image of dM under the transformation log(z - c) for a certain c e dM ? Note the cusp on the main cardioid in the upper right; looking to the left in the figure corresponds to

zooming in towards the point c. (Namely, c = -0.39054087... - 0.58678790i... the point on the boundary of the main cardioid corresponding to the golden mean Siegel disk.). Note the cusp on the main cardioid in the upper right; looking to the left in the figure corresponds to zooming in towards the point c. "[14]

• "Legendary side scrolling fractal zoom. 1 Month + (Interpolator+Video Editor) = Log(z). This means logarithmic projection for this location, that gives this interesting side-scrolling plane ^^)﻿"[15]
• " There are no program that can render this fractal on log(Z) plane. But you can make it in Ultra Fractal or in similar software with programmable distributive. Formula is:C = exp(D), for D - is your zoomable coordinates﻿" SeryZone X

# point c description

• c value
• Cartesion description
• real part
• imaginary part
• polar description:
• (external or internal ) angle
• ( external or internal) radius

# Point Types

point =pixel of parameter plane = c parameter

## Criteria

Criteria for classification of parameter plane points :

• arithmetic properties of internal angle (rotational number) or external angle
• in case of exterior point:
• type of angle : rational, irrational, ....
• preperiod and period of angle under doubling map
• in case of boundary point :
• preperiod and period of external angle under doubling map
• preperiod and period of internal angle under doubling map
• set properties ( relation with the Mandelbrot set and wakes)
• interior
• boundary
• exterior
• inside wake, subwake
• outside all the wakes, belonging to a parameter ray landing at a Siegel or Cremer parameter,
• geometric properities
• number of of external rays that land on the boundary point : tips ( 1 ray), biaccesible, triaccesible, ....
• position of critical point with relation to the Julia set
• Renormalization

## Classification

There is no complete classification. The "unclassifed" parameters are uncountably infinite, as are the associated angles.

### Simple classification

• exterior of Mandelbrot set
• Mandelbrot set
• boundary of Mandelbrot set
• interior of Mandelbrot set
• centers,
• other internal points ( points of internal rays )

### partial classification of boundary points

Classification :[16]

• Boundaries of primitive and satellite hyperbolic components:
• Parabolic (including 1/4 and primitive roots which are landing points for 2 parameter rays with rational external angles = biaccesible ).
• Siegel ( a unique parameter ray landing with irrational external angle)
• Cremer ( a unique parameter ray landing with irrational external angle)
• Boundary of M without boundaries of hyperbolic components:
• non-renormalizable (Misiurewicz with rational external angle and other).
• renormalizable
• finitely renormalizable (Misiurewicz and other).
• infinitely renormalizble (Feigenbaum and other). Angle in turns of external rays landing on the Feigenbaum point are irrational numbers
• non hyperbolic components ( we believe they do not exist but we cannot prove it ) Boundaries of non-hyperbolic components would be infinitely renormalizable as well.

Here "other" has not a complete description. The polynomial may have a locally connected Julia set or not, the critcal point may be rcurrent or not, the number of branches at branch points may be bounded or not ...

Examples: