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 $T^0(C), T^1(C), T^2(C), \ldots T^n(C)$ .¹T^n(C)[/latex] 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 $n>0$ .] These circles are called the forward iterates of circle $C$ under $T$ . Remarkably, the image of any circle is a circle.²T(z)=az[/latex] is a special case of the Mobius transformation. Every Mobius transformation maps circles to circles, where circle broadly includes both circles and lines.]

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|a|<1[/latex], iterates would approach the origin under $T$ and so the origin and infinity would be the sink and source respectively.]

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.

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These are called elliptic transformations. ↩

Michael Laszlo

It is not a something, but not a nothing either

Loxodromic transformations