Ellis Caird 
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The Calkin-Wilf Tree

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Rational Thoughts

I recently found myself needing to understand the dynamics of the following two rational functions during my research.1 Consider the suggestively named maps L,R ⁣:Q→QL,R\colon \mathbb{Q} \to \mathbb{Q} defined via

L(q)=q1+q,R(q)=q+1.L(q) = \frac{q}{1+q},\quad R(q) = q+1.

Perhaps even more suggestively we could let q=abq = \frac{a}{b} and rewrite the definition as

L(ab)=aa+b,R(ab)=a+bb.L\left(\frac{a}{b}\right) = \frac{a}{a+b},\quad R\left(\frac{a}{b}\right) = \frac{a+b}{b}.

Have a play! Use the arrow keys in the sketch below to apply combinations of L and R to 1/1. Press r to reset.

Rational Questions

If we can pry ourselves away from mashing left and right then we might find that we have a couple of natural questions.

  • Starting from 1, which rationals can we obtain using combinations of L and R?
  • Given a rational that can be obtained from 1, how many ways are there to do so?

Footnotes

  1. For anyone interested, I’m currently trying to understand the structure of the module category of a tubular algebra. Regular modules have a notion of ‘slope’ which is just some rational number and determines a whole tubular family. Shrinking functors in the sense of Ringel allow you to pass between tubular families. They thus also act on slopes and the maps L and R are the induced maps for my two particular shrinking functors. ↩

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