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 :)).

Martin Ender
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    The answers to [this question](http://math.stackexchange.com/questions/1681993/why-is-1-1-1-1-1-not-real) give a lot of insight. – Ritz Mar 09 '16 at 15:24
  • This is pretty cool. Are there specific questions that you can articulate? – abiessu Mar 09 '16 at 15:24
  • @Ritz: that question deals with $x\to 1+\frac 1x$, which is similar, but not quite the same... – abiessu Mar 09 '16 at 15:26
  • @abiessu My specific question is really "has this been studied/does this have a name/are there well-known properties I could find somewhere?" While I do have several open questions I'd like to answer about the map, this question isn't about getting help with the actual analysis, although I might post a separate one in the future if I keep investigating this myself. (I hope questions like this one are on-topic here.) – Martin Ender Mar 09 '16 at 15:26
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    @Ritz You are aware that this is my own question you linked? ;) – Martin Ender Mar 09 '16 at 15:27
  • @MartinBüttner Oh, my bad, I overlooked both the username and the fact that $z_{n + 1} = z_n - \frac{1}{z_n}$ is very different from $z_{n + 1} = 1 - \frac{1}{z_n}$. – Ritz Mar 09 '16 at 15:29
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    Section 1.9 of Alan Beardon's [Iteration of Rational Functions](http://www.springer.com/us/book/9780387951515) 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. – Mark McClure Mar 09 '16 at 17:20
  • @MarkMcClure Thanks, I'll try to see if I can get my hands on a copy of that. I actually did notice the swapped roles of the imaginary and real axes between $z - 1/z$ and $z + 1/z$ but somehow I didn't grant that much significance. – Martin Ender Mar 09 '16 at 17:25
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    @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. – Martin Ender Mar 09 '16 at 20:28
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    @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! – Mark McClure Mar 09 '16 at 21:27
  • if you only look at $\Bbb R \cup \{ \infty \}$ then your dynamical system is isomorphic to $z \mapsto z^2$ on the unit circle. What's funny is that the isomorphism is not analytic (idk if it's even differentiable), and that the respective analytic continuations to the Riemann sphere look very different. – mercio Mar 31 '16 at 14:30
  • Not sure if this is a sufficient answer so adding as a comment and am surprised nobody mentioned it yet. The function $f(z) = z + \frac{1}{z}$ is the [Joukowski][1] function which finds use in fluid flows and modelling aerofoils. Some description of recursion with this formula is described [2] [1]:http://mathfaculty.fullerton.edu/mathews/c2003/JoukowskiTransMod.html [2]: http://math.coe.uga.edu/olive/Joukowski.Web/Joukowski.Paper.html – PM. Feb 27 '17 at 22:08
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    Your map can be viewed as a Newton's method iteration, using the function $f(z):=\exp(z^2/2)$ (or any nonzero multiple of it). – user1337 Mar 14 '17 at 06:42
  • @MartinEnder A generalisation of your map would be quadratic rational maps, after Beardon's Iteration of Rational Functions, there is also Milnor's [Remarks on Quadratic Rational Maps](https://arxiv.org/abs/math/9209221) or [here](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.em/1062620736) – mdave16 Mar 14 '17 at 22:31
  • I would comment with more literature, but I think that one will keep you busy for a few weeks. Milnor is rather dense to read if you are new to the area. And can be rather tiring even to those who know their way around. – mdave16 Mar 14 '17 at 22:35
  • Have you found an explicit formula yet? And what conjectures about it are you trying to prove. – Franklin Pezzuti Dyer Apr 29 '17 at 21:24
  • @FranklinP.Dyer I haven't. I guess the conjectures I'm most interested in are whether the number of orbits of a given period is really A001037 and whether there are any closed form expressions relating to the geometric shape of the fractal (e.g. a curve touching all "circles"). – Martin Ender Apr 29 '17 at 21:50
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    Here's something interesting and possibly related: if you let $q_{n+1}=\frac{1}{2}q_n-\frac{1}{2q_n}$, then $q_n$ is given by $$\tan(2^n\tan^{-1}x+\pi \frac{2^n-1}{2})$$ – Franklin Pezzuti Dyer Apr 30 '17 at 20:57
  • @Ritz and Martin Ender: linking to another question by the same person is still good, because we are not replying only to that one person, so others will benefit from the link. Actually it would be good practice to include that upfront, if/when you are aware that the questions are linked. – Rolazaro Azeveires Dec 24 '17 at 11:31
  • @RolazaroAzeveires I'm aware, but my other question didn't seem particularly relevant to this one. The recurrence looks similar, but is actually quite different. – Martin Ender Dec 24 '17 at 11:36
  • @MartinEnder sure, that was I meant (but did not explain in full, I see) with the "if": if you think they are connected, off course you may think not, while others may think they are. There was no intention of negative criticism to you in my comment, only a note clarifying to future readers that Ritz's link was somehow funny, but not wrong, quite the opposite. – Rolazaro Azeveires Dec 24 '17 at 11:40

1 Answers1


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.

kimchi lover
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  • Why isn't the "Boole map" listed in Wikipedia's list of chaotic maps? –  Dec 09 '18 at 21:36
  • @EulerSpoiler Maybe it is, but not under that name? Or, maybe that list is kind of shlocky, the scrapings off of Chaos Blog etc. – kimchi lover Dec 09 '18 at 23:37