Loxodromic transformations

A transformation that rotates and expands is said to be loxodromic. Start with a circle near the origin and apply an origin-centered loxodromic transformation to it again and again, and the circle will spiral out from the origin. The following program explores the dynamics of the transformation T(z) = az. As in the previous program, the complex number a is fixed by the abs and arg sliders, which set |a| and \theta_a. In particular, when |a|>1 and \theta_a\neq0, the transformation is loxodromic.

Dynamics of a circle

Start with a circle C centered on the real axis, whose position and radius are fixed with the re and radius sliders. Then iteratively apply the transformation T(z) = az to obtain the sequence of circles T^0(C), T^1(C), T^2(C), \ldots T^n(C).1 These circles are called the forward iterates of circle C under T. Remarkably, the image of any circle is a circle.2

The program also produces the backward iterates T^{-1}(C), T^{-2}(C),\ldots T^{-n}(C). As n increases, we see the forward and backward orbits of circles emerging from the seed circle C.

If |a|>1 and \theta_a\neq0, forward iterates spiral away from the origin and backward iterates spiral toward the origin. If \theta_a=0, forward iterates head straight away from the origin and backward iterates head straight toward the origin. In both cases (i.e., where |a|>1), there are two fixed points: the origin, called the source, and infinity called the sink. 3

There is neither expansion nor contraction when |a|=1, and the circles orbit the origin at fixed distance assuming \theta_a\neq 0.4 In this case, the origin and infinity are considered neutral fixed points.

  1. Here T^n(C) applies T n times to circle C. That is, T^0(C) = I is the identity transformation, and T^n(C)=T(T^{n-1}(C)) for all n>0. ↩
  2. As defined, T(z)=az is a special case of the Mobius transformation. Every Mobius transformation maps circles to circles, where circle broadly includes both circles and lines. ↩
  3. If |a|<1, iterates would approach the origin under T and so the origin and infinity would be the sink and source respectively. ↩
  4. These are called elliptic transformations. ↩