Angles and Rays of M and J sets

Angles and Rays of M and J sets

Point of exterior of M-set has one external angle.
M-set
- Point of boundary of M-set has external angle ( one or more) and internal angle ( only one ).
  Boundary point which has 2 external angles ( 2 external rays land on this point) is called biaccessible. It is a root of component.
- Point of M-set of interior of M-set has an internal angle ( only one).

Of course every point of 2D plane ( complex number) has another angle = Argument.

So we have 4 types of angles:

complex angle of complex point := Argument ( c)
last point phase of orbit = Argument ( Zn) where Zn := F_J( Z0=0,n,c ) it is used in decomposition of exterior of M-set
external angle of Mandelbrot set:= Arg _M(c) = Arg ( Phi _M ( c) ) where Phi _M it is a Boettcher map
internal angle of hyperbolic component of Mandelbrot set := Argument ( multiplier (c) )

Angle is a real number.
angle = Argument ( complex_number)
Argument ( c) : C --> R
because :
c := Re(c) + Im (c)* i =
Log(c) := Ln ( r ) + i * ( angle + 2 * Pi* k )
Argument(c) := Im ( Log ( c) )

Angles can be measured in degrees, radians and turns.
360 [degrees] = 2 * Pi [radians] = 1 [ turn]
angles are measured in counterclockwiserotation
and modulo 1
it means 0,2 = 1,2 mod 1 = 2,2 mod 1

Forms of angles:

binary form ( sequence of binary digits)
decimal form :
- vulgar fraction = rational numbers
  - irreducible (or in lowest terms) fraction
  - external angles can be in normalized form : numerator / (2^p-1)
    where p is the period of hyperbolic component
    form n/d where d=(2^p-1) is normalized form
    form n/(2^p-1) is explicit normalized form
    "if the denominator is 2^p-1, the external angle leads to the root of a mu-atom of period p.
    All other denominators correspond to external angles that lead to branch points, tips, etc. The external angles with powers of 2 in the denominators correspond to the bifurcation in rotational symmetry of the filaments surrounding all islands. This also shows up in the symmetry of embedded Julia sets and their paramecia." from External Angle Robert P. Munafo

Ray is a set of points with the same (external or internal) angle.
It is a curve, sometimes a line.
External ray of Mandelbrot set is called parameter ray because it is on parameter plane.
External ray of Julia set is called dynamic ray because it is on dynamic plane.
R(c,angle) = { c | Arg _M(c) = angle }
Rays are labeled by the coresponding angles.

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Autor: adammaj1-at-o2-dot--pl
Adam Majewski
Feel free to e-mail me. (:-))

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