A while ago I stumbled upon an interesting recurrence of the form $$x_{n+1} = {\rm e}^{\jmath \cdot \alpha \cdot x_n}.$$

I studied it for positive, real-valued $\alpha$ though one could look also at complex values. I noticed that for a range of $\alpha$, this recurrence seems to have one stationary point. Trivially, for $\alpha=0$ it is $x_\infty = 1$ but I am more interested to compute it for other values of $\alpha$. I couldn't find anything helpful analytically (there is a lot on linear recurrence relations but I didn't find anything helpful on nonlinear ones). Therefore, I continued to study it numerically. Here is what I found:

  • For $0 \leq \alpha \lessapprox 1.96$ there seems to be a single stationary point. Here is a plot of the numerical values I found, displaying $\Re(x_\infty)$ and $\Im(x_\infty)$ for various values of $\alpha$.Stationary points found numerically
  • For each $\alpha$ the sequence seems to converge to the fixed point regardless of the initialization $x_0$, but I'm not sure of it. The problem when you check this numerically is that some very large values can occur. E.g., for $\alpha = \pi/2$ and $x_0 = 3-7j$ we have:

\begin{align} x_1 & = \exp[7\pi/2]\cdot \exp[\jmath 3\pi/2] = \exp[7\pi/2]\cdot(-\jmath) \approx - 59610\jmath \\ x_2 & = \exp[-\jmath \pi/2\cdot \jmath \cdot \exp(\jmath 7\pi/2)] =\exp[\exp(7\pi/2)] \approx {\rm e}^{59610} \\ x_3 & = \exp[\jmath \pi/2 \exp(\exp[7\pi/2])] = \exp[\jmath \varphi] \end{align}

where $\varphi$ (modulo $2\pi$) is not very easy to compute but $x_3$ is small, since $|x_3|=1$. From this point on it converges. If you try to plot this numerically, you will get fractal-looking shapes like these:

Convergence vs. initial point Convergence vs. initial point, zoomed

Here, the blue color indicates the points where $x_n$ became to large to fit into my float data type (i.e., I got INF) which could mean either divergence or just numerical problems like the one I explained above. The other colors indicate convergence after $n$ iterations, defined as getting within a radius of 0.001 of the single stationary point found earlier.

  • For $\alpha>1.962$, the sequence seems to form limit cycles of a discrete number of points but these are even harder to predict and seem somewhat chaotic.

Now here are my questions:

  1. Is it possible to compute $\lim_{n\rightarrow \infty} x_n$ analytically?
  2. The recurrence seems to be a quite generic one, does it have a name under which I could search further?
  3. Is anything known on the convergence? Does it have a nontrivial region of convergence and is it possibly really fractal? Or are the fractal looking images merely an artifact of my limited data type?
  • 1
    $\begingroup$ You are generating Julia sets of exponential functions, which you can read more about on Bob Devaney's list of publications. $\endgroup$ May 5, 2019 at 12:18
  • $\begingroup$ Cool! Finally an interesting clue to unveil the fractal nature of these recurrence relations. Thanks so much! $\endgroup$
    – Florian
    May 6, 2019 at 8:22


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