The Unit Circle Group is a subgroup of the group of Möbius Transformations. An elementof this group has the form T(z) =

with |a|² - |b|²  = 1.

If a = 1, then T is just the identity map; if |a| = 1, then b= 0 and T is a rotation through twice the angle Arg(a), ( 0 ≤ Arg(a) < π). In this case the only fixed point is the origin. If Arg(a) is a rational multiple of 2π then T has periodic points, otherwise T is chaotic. If =a and b are real, a >1  and b > 0, then 1 and -1 are fixed points of T. -1 is a “source” and +1 is a “sink”, that is, T  moves any point inside the circle  away from -1 and closer to +1. If |a| > 1, |b| is determined. If Arg(a) and/or Arg(b) are not zero, then points will be rotated as well as stretched.

Now we will construct a design made up of disjoint circles placed inside the unit circle and illustrate the dynamics of applying transformations from the Unit Circle Group by “continuously” varying the parameters |a|, arg(a) and arg(b). (Since |b|²  = |a|² – 1, there are only 3 parameters that can be freely chosen).

Original design

One iteration: a = 1, b = 0

 

One iteration: a = 1.25, arg b = 0

One iteration |a|= 1.25, arg(a) =π/4, arg(b) = 0

One iteration |a|= 1.25, arg(a) =π/8, arg(b) = 0

 

 

After 4 iterations; a = 1, b = 0	4 iterations: a = 1.25, arg a = arg b = 0
4 iterations |a|= 1.25, arg(a) =π/4, arg(b) = 0	4 iterations:|a|= 1.25, arg(a) =π/8, arg(b) = 0

 

More examples of circle pictures
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