Has anyone studied the real function $$ f(x) = \frac{ 2 + 7x - ( 2 + 5x )\cos{\pi x}}{4}$$ (and $f(f(x))$ and $f(f(f(x)))$ and so on) with respect to the Collatz conjecture?

It does what Collatz does on integers, and is defined smoothly on all the reals.

I looked at $$\frac{ \overbrace{ f(f(\cdots(f(x)))) }^{\text{$n$ times}} }{x}$$ briefly, and it appears to have bounds independent of $n$.

Of course, the function is very wiggly, so Mathematica's graph is probably not $100\%$ accurate.

  • $\begingroup$ Please review the formulas because I'm not sure if that's exactly what you wanted to write. $\endgroup$ – Adrián Barquero Apr 2 '11 at 19:32
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    $\begingroup$ @Adrian Thanks for the cleanup; yes, that's what I meant. $\endgroup$ – barrycarter Apr 2 '11 at 19:54

The analogue of the Collatz conjecture would be false, since the image of an interval of length $1$, say, contains an interval of length $1$ strictly to the right. That means you can find a point whose images keep moving to the right.

  • $\begingroup$ Nice observation. I don't understand your conclusion, though. I can see that for every $n$, you can find a point that moves to the right $n$ times, but how do you find a point that keeps moving (and doesn't reach a limit, either)? $\endgroup$ – Sebastian Reichelt Apr 2 '11 at 22:01
  • $\begingroup$ Here is something to consider, though: There are lots of $x$ that map to themselves, the lowest around $1.736$. I guess that does make the analogue of the Collatz conjecture false, whatever that analogue might be exactly. $\endgroup$ – Sebastian Reichelt Apr 2 '11 at 22:14
  • $\begingroup$ You use the nested interval theorem: A nested sequence of closed intervals has nonempty intersection. $\endgroup$ – Douglas Zare Apr 2 '11 at 23:07

Yes, it has been studied in

Xing-yuan Wang and Xue-jing Yu (2007), Visualizing generalized 3x+1 function dynamics based on fractal, Applied Mathematics and Computation 188 (2007), no. 1, 234–243. (MR2327110).

I have found this reference in Jeffrey Lagarias's "The $3x + 1$ Problem: An Annotated Bibliography, II (2000-2009)".

This real/complex interpolation of yours is not the only one imaginable, you will find several others either in Lagarias' article cited above, or in the first part of this series of articles, namely in "The $3x+1$ problem: An annotated bibliography (1963--1999)".


Yes people have studied that, as well as extending it to the complex plane, e.g.:

Chamberland, Marc (1996). "A continuous extension of the 3x + 1 problem to the real line". Dynam. Contin. Discrete Impuls Systems. 2 (4): 495–509.

Letherman, Simon; Schleicher, Dierk; Wood, Reg (1999). "The (3n + 1)-Problem and Holomorphic Dynamics". Experimental Mathematics. 8 (3): 241–252. Cached version at CiteSeerX

You'll also find something about that in the corresonding wikipedia article, including the Collatz fractal.


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