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I have recently been playing around with the discrete map

$$z_{n+1} = z_n - \frac{1}{z_n}$$

That is, repeatedly mapping each number to the difference between itself and its reciprocal. It shows some interesting behaviour. This map seems so simple/obvious, I highly doubt this has never been analysed before. However, I (and several people I asked) have been unable to turn up any form of literature on it online (partly because it's hard to google maths and partly because a lot of people on the internet ask about "the difference between 'inverse' and 'reciprocal'"). I also couldn't find it in Wikipedia's List of chaotic maps. Does this map have a name I could look up?

I am vaguely aware of Möbius transformations, but clearly this isn't one (although it might be possible to express it as a combination of two or more Möbius transformations).

I'm listing here some things I have observed about the map (some proven, some conjectured), in case they ring any bells for people in terms of related maps or generalisations:

  • The map is chaotic on the real line. All orbits are unstable, and the map is highly sensitive to initial conditions. Here are the first 1000 iterations starting from each of $4.000001$ to $4.000009$:

    enter image description here

  • It's not chaotic anywhere else in the complex plane: all trajectories that don't start on the real line will eventually be attracted by the imaginary axis and increase in magnitude without bound.

  • Before doing that, any trajectory can jump in and out of the unit circle arbitrarily often. If I colour the complex plane depending on the "inside-outside" pattern of the trajectory, I get a nifty fractal of deformed circles covering the real axis (repeated colours are due to a limited palette):

    enter image description here

  • I believe that the number of orbits of period $n$ is given by OEIS A001037 (all of which are on the real line), provided you count the fixed points $\pm \infty$.

  • As pointed out by Mark McClure in the comments, the map $$ z_{n+1} = z_n + \frac{1}{z_n} $$ is identical to the one I'm looking at, but with the roles of the real and imaginary axis swapped. This map, as well as $z_{n+1} = 2z - 1/z$, has a short section in Alan F. Beardon's Iteration of Rational Functions, but that doesn't go far beyond what I've mentioned above, and doesn't help me in finding further literature about the maps at this point.

Whether this has any use or not, analysing this map is a nice exercise in recreational maths for me, but I'm somewhat reaching the limits of my capabilities and would like to find out if anything else is known about this map (or whether I'll have to prove my conjectures myself :)).

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    $\begingroup$ The answers to this question give a lot of insight. $\endgroup$
    – Ritz
    Mar 9, 2016 at 15:24
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    $\begingroup$ @Ritz You are aware that this is my own question you linked? ;) $\endgroup$
    – Martin Ender
    Mar 9, 2016 at 15:27
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    $\begingroup$ Section 1.9 of Alan Beardon's Iteration of Rational Functions examines the iterates of $R(z)=z+1/z$. Your function $f(z)=z-1/z$ is dynamically conjugate to $R$ via the (non-analytic, but rigid) conjugacy $\varphi(z)=i\,\bar{z}$. As a result, the dynamical behavior is virtually identical. In particular, the Julia set of $f$ is the whole real line, as you've rather discovered experimentally. $\endgroup$
    – Mark McClure
    Mar 9, 2016 at 17:20
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    $\begingroup$ @MarkMcClure I managed to find a copy in the library (although it's section 1.7 in my edition ;)). Section 1.8 also seems relevant ($R(z) = 2z - 1/z$, and I was thinking that $az - b/z$ would be the natural generalisation of my maap), but they're both pretty short. This looks interesting though, I'll try to read a bit more of the book. Unfortunately, while this is starting point, I don't think it helps me in finding any further analysis of the maps. $\endgroup$
    – Martin Ender
    Mar 9, 2016 at 20:28
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    $\begingroup$ @MartinBüttner Yes, I can believe that. The general area that you're experimenting in, complex dynamics, is spectacular stuff but has a bit of a learning curve. The specific example you've chosen is particularly tricky, since there's no attractive behavior at all - just a neutral fixed point at $\infty$. A good answer should demonstrate the conjugacy between your $f$ and Beardon's $R$ and describe the broader implications. That's why I chose to leave a comment, rather than an answer, which I don't quite have the time for at the moment. I do hope you have fun with it, though! $\endgroup$
    – Mark McClure
    Mar 9, 2016 at 21:27

1 Answer 1

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This is the "Boole map", described by George Boole (yes, that George Boole) in his 1857 "On the Comparison of Transcendents, with Certain Applications to the Theory of Definite Integrals". A standard modern reference is the 1973 paper by R. Adler and B. Weiss, "The ergodic measure preserving transformation of Boole". It shows up in the answer to this MSE question.

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  • $\begingroup$ Why isn't the "Boole map" listed in Wikipedia's list of chaotic maps? $\endgroup$
    – user584285
    Dec 9, 2018 at 21:36
  • $\begingroup$ @EulerSpoiler Maybe it is, but not under that name? Or, maybe that list is kind of shlocky, the scrapings off of Chaos Blog etc. $\endgroup$
    – kimchi lover
    Dec 9, 2018 at 23:37

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