This is now entirely a semantic argument. I would argue "physics" and "nature" are almost interchangeable here. KAM tori naturally occur in ocean currents.
Critical phenomena, like phase transitions, are essentially defined by fractal like properties. Scale invariance in systems near a phase transition inspired the technique now called renormalization group. It remains a central tool in statistical physics, but isn't really in vogue at the moment. Kenneth Wilson won a Nobel prize in 1982 for his work on RG.
Some very cool physics involved-- a lot of basic quantum mechanics (electrons hopping on a lattice), but it has a lot of really broad implications about symmetries. There is a lot of recent interest in the butterfly [1,2].
It turns out that it is a very important state to study the Quantum Hall effect[3], and the fractional Quantum Hall effect[4], which seem to be in vogue in Condensed Matter physics these days.
Sorry to bog you down with details, I just wanted to mention how neat it is that Hofstadter's butterfly is still being studied today!
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[ 4.7 ms ] story [ 24.7 ms ] thread[1] https://en.wikipedia.org/wiki/Renormalization_group
http://en.wikipedia.org/wiki/Almost_Mathieu_operator
It turns out that it is a very important state to study the Quantum Hall effect[3], and the fractional Quantum Hall effect[4], which seem to be in vogue in Condensed Matter physics these days.
Sorry to bog you down with details, I just wanted to mention how neat it is that Hofstadter's butterfly is still being studied today!
[1] http://www.nature.com/nature/journal/v497/n7451/full/nature1... [2] http://physics.aps.org/articles/v6/118