Limited Mandelbar Sets

step

6

threshold

2

exponent

2

conjugation
mesh

Consider the mapping

f

c

:z↦

2

z

+c

. The Mandelbrot set consists of those complex numbers

c

such that the iterates of

(n)

f

c

(0)

do not tend to infinity as

n∞

. Points with an iterate greater than 2 in absolute value diverge and are thus excluded from the Mandelbrot set.

Now consider taking the complex conjugate

z

and arbitrary powers

n

z

until the result exceeds an arbitrary value w. The mapping is then

f

c

:z↦

n

z

+c

. The resulting sets are called Mandelbar sets because of the bar denoting conjugation.

The image plots the region of points having iterates with norms less than w.

Michael Schreiber