Classroom Notes: Images of Fixed-Point Continued Fractions
John Gill December 2013
Abstract: Computer generated images in the complex plane of iterations of certain analytic continued fractions.
Fixed-point continued fractions have the form
(1)
α1 β1
α2 β 2
… where α k , β k are the fixed points of the function
α1 + β1 − α2 + β2 −
f k (ζ ) =
αk βk
. The nth approximant of the continued fraction then can be written
αk + βk − ζ
Fn(ζ ) = f1 f2 f n(ζ ) . If α k = α k ( z ) and β k = β k ( z ) , we have
Fn( z ,ζ ) = f1 f2 f n( z ,ζ ) . The following images show magnitudes of simple complex flux
(SCF), Fn( z ,0) , or fixed-point flux (FPF), Fn( z ,0) − z . Dark=small, light=large. Several
variations on FPCFs are shown, as well. [Liberty Basic V 4.04 (2013)]
Example 1: f k ( z ,ζ ) =
kz
, −6 ≤ x , y ≤ 6 , n = 10
k + z −ζ
SCF & FPF
2
(1 + ik )
2
z
(.5 + ik ) z , −6 ≤ x , y ≤ 6 , n = 10 SCF
Example 2: f k ( z ,ζ ) =
and f k ( z ,ζ ) =
1 + ik + z − ζ
.5 + ik + z − ζ
A Variation on fixed-point CFs.
Example 3:
f k ( z ,ζ ) =
Note: a programming error produced these two images!
kz(1 − kz )
, −6 ≤ x , y ≤ 6 , n = 10 SCF
1−ζ
Example 4:
f k ( z ,ζ ) =
Example 5: f k ( z ,ζ ) =
kz(1 − kz )
, α k =1+ik , −6 ≤ x , y ≤ 6 , n = 10 SCF
αk z
1−
αk + z − ζ
(1 − ik ) z
1 − ik + z − ζ
, −7 ≤ x , y ≤ 7 , n = 10 SCF
Example 6: f ( z ,ζ ) =
z
, −6 ≤ x , y ≤ 6 , n = 10 SCF. Another variation on FPCFs.
k + z −ζ
Example 7: f k ( z ,ζ ) =
kz 2
, −10 ≤ x , y ≤ 10 , n = 10 SCF. Another variation on FPCFs.
k + z −ζ
Example 8: f ( z ,ζ ) =
(1 + 2ki )2 z
, −10 ≤ x , y ≤ 10 , n = 10 SCF. Another variation on
1 + 2ki + z − ζ
FPCFs.
Example 9: f ( z ,ζ ) =
i(1 + ik )z
, −8 ≤ x , y ≤ 8 , n = 10 SCF. Variation on FPCFs.
1 + ik + z − ζ