The Imaginary Power Tower: Part II

This is a continuation of last week’s post: LINK

The complex power tower is defined by an `infinite tower’ of exponents:

$\displaystyle Z(z) = {z^{z^{z^{.^{.^{.}}}}}} \,.$

The sequence of successive approximations to this function is

$z_0 = 1 \qquad z_{1} = z \qquad z_{2} = z^{z} \qquad \dots \qquad z_{n+1} = z^{z_n} \qquad \dots$

If the sequence ${\{z_n(z)\}}$ converges it is easy to solve numerically for a given ${z }$ .

Pursuit-triangle In Part I we described an attempt to fit a logarithmic spiral to the sequence ${\{z_n(i)\}}$ . While the points of the sequence were close to such a curve they did not lie exactly upon it. Therefore, we now examine the asymptotic behaviour of the sequence for large ${n}$ .

Let us consider the specific case ${z = i}$ and suppose that ${z_n = (1+\epsilon)Z}$ where ${\epsilon}$ is a small complex number (we note that ${\epsilon}$ depends on ${n}$ ). Then we find that ${z_{n+1} = Z^\epsilon\cdot Z}$ so that

$\displaystyle \left( \frac{z_{n+1}-Z}{z_n-Z}\right) = \left( \frac{Z^\epsilon-1}{\epsilon}\right) \,.$

By L’Hôpital’s rule, the limit of the right-hand side as ${\epsilon\rightarrow0}$ is ${\log Z}$ . Thus for small ${\epsilon}$ , or large ${n}$ , we have

$\displaystyle ( z_{n+1}-Z ) \approx \log Z \cdot (z_n-Z)$

and the sequence of differences ${\{z_{n+k}-Z\}}$ lies approximately on a logarithmic spiral

$\displaystyle z_{n+k} \approx Z + (\log Z)^k \cdot (z_n-Z) \,. \ \ \ \ \ \ \ \ \ \ (1)$

The Asymptotic Spiral

Sequence {z_n} starting at n=30, with logarithmic spiral (1) superimposed.

In the Figure above, we show the sequence ${\{z_n(i)\}}$ for ${n\ge 30}$ . The points spiral around the limit point ${Z(i) = (0.438283, 0.360592)}$ converging towards it. The logarithmic spiral defined above is superimposed. It provides an excellent fit to the points of the sequence.

In the Figure below, we show the same sequence of points. The points ${z_n}$ fall clearly into three distinct sets, each in the form of a spiral. Three logarithmic spirals are superimposed on the plot. The three sets of points lie on or very close to these spiral arms. For ${z=i}$ the value of ${\log Z}$ is ${\log Z \approx (-0.566+0.688)}$ so that ${|\log Z| = 0.892}$ and ${\arg(\log Z) = 129^\circ}$ . Thus each third point is shifted through slightly more than a full rotation: ${3\times 129^\circ = 360^\circ+27^\circ}$ . This explains the separation of the terms into three sets, each falling on a distinct spiral arm.

Points {z_n} for n>=30, with three supernumerary spirals superimposed.

From the asymptotic analysis, it is clear that the occurrence of these three “supernumerary spirals” is not an accident. The appearance of such spirals is familiar in many contexts. For example, in the seeds of a sun-flower, clockwise and anti-clockwise spirals are evident. By changing the parameter ${z}$ it should be possible to tune the limit ${Z(z)}$ to have spirals of a particular shape. Patterns of this sort are also found in pursuit problems: If three ships initially at the vertices of an equilateral triangle follow courses such that each bears towards its counter-clockwise neighbour, three spiral arms are traced out (see Figure at head of this post).

Finally, we raise a question: What is the domain of convergence of the complex power-tower? Is the boundary of this domain regular or fractal? We hope to return to this question in a later post. NOW DONE

Sources

${\bullet}$ The Power Tower. Blog post on www.thatsmaths.com.

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