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Let $I$ be the unit interval and $f, g:I\to I$ be continuous functions. Assume that $f\circ g = g\circ f$.

This post shows that if $f$ and $g$ in addition are assumed to be increasing then $f$ and $g$ have a common fixed point.

Question. Does the same conclusion hold without assuming that $f$ and $g$ are monotonically increasing?

I couldn't construct any examples where the same conclusion does not hold.

Some Observations.

  1. If $\text{Fix}(f) = \{x\in I: f(x) = x\}$ and $\text{Fix}(g) = \{x\in I:\ g(x) = x\}$, then $\text{Fix}(g)$ is $f$-invariant and $\text{Fix}(f)$ is $g$-invariant.

  2. The set $\{x\in I:\ f(x) = g(x)\}$ (which, as shown here, is not empty) is both $f$ and $g$-invariant.

caffeinemachine
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  • A really interesting question. Many researchers have studied such questions but, as far as I can remember, they always assumed homeomorphisms of the unit interval. In the 2nd question that you linked they don't mention monotonicity. Or am I missing anything? – Kurt G. Mar 27 '22 at 08:44
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    This is a hard question which was open for a long time. The answer is "no": Boyce, W.M.: Commuting functions with no common fixed point. Trans. Amer. Math. Soc. 137, 77-92 (1969). – Gerd Mar 27 '22 at 09:03
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    @KurtG. Yes one does not require monotonocity for the second observation. – caffeinemachine Mar 27 '22 at 09:14
  • Very interesting ! Would like to point out that for monotone functions that commute there are a number of interesting results. For example M.K. Fort, showed in The embedding of homeomorphisms in flows Proc. Amer. Math. Soc., 6, 960–967, (1955) that there is a one paramter semigroup of homeomorphisms which $f$ and $g$ belong to. This is also related to a functional equation that goes back to Niels Hendrik Abel. – Kurt G. Mar 27 '22 at 09:21
  • @Gerd Thank you so much for the reference. – caffeinemachine Mar 27 '22 at 10:52

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