# Who first proved ergodicity of irrational rotations of the circle?

It is a classic result that the irrational rotations of the circle are ergodic. Formally, let $T:\mathbb{T}\to \mathbb{T}$ be defined by $Tz=ze^{2\pi i\alpha}$. If $\alpha$ is irrational, then $T$ is ergodic. This result appears in many textbooks (e.g., Walters, An Introduction to Ergodic Theory), and even in Wikipedia. However, none of these refer to the original. A Google Scholar search didn't help either.

My question is simply, who was the first to prove or to notice this result, and is there a reference to the original paper?

• The modern proof is due to Weyl and others in 1910. en.m.wikipedia.org/wiki/Equidistribution_theorem – Anthony Quas May 16 '17 at 8:47
• It seems wrong. Unless I don't properly understand the concept of "ergodic", this fails for $\pi$, which is irrational, and for its integer and fractional multiples. Rather, rational values of $\alpha$ should be expected to lead to ergodic rotations, no? – Kaz May 17 '17 at 17:45
• Did you perhaps mean $Tz = ze^{2\pi i\alpha}$? So that 90, 180, 45 or 360 degree rotations are considered rational degree rotations are considered rational, corresponding to rational values of $\alpha$? – Kaz May 17 '17 at 17:47
• Amir, I think it's counterproductive to leave accepted such a questionable answer. – YCor Nov 10 '17 at 16:19
• @AmirSagiv I think that the accepted answer (Carlo's) is not a good one because there it has serious criticism in its comments (esp. Alexandre's comment), and no attempt to respond to this criticism. Not to accept this answer does not mean accept another one: my comment did not refer to any other answer. Carlo claims that "the proof goes back to Oresme" with only a few references. The last one refers to ergodicity in its title, but this is not enough to be convincing. I could also claim it's Archimedes and refer to the complete works of Archimedes... – YCor Nov 12 '17 at 15:45

the proof goes back to Nicole Oresme in his paper De commensurabilitate vel incommensurabilitate motuum celi [On the Commensurability or Incommensurability of the Motions of the Heavens], dated around 1360, see

• Thanks! I guess though that Oresme didn't have the harmonic-analysis tools used in Ergodic thery, do you know who introduced the modern proof? – Amir Sagiv May 16 '17 at 8:28
• The middle bullet links to a paper by Jan von Plato, which ends with the following sentences: "In reading Oresme it at times appears as if he thought the incommensurable conjunctions would be equally distributed in all directions. Such equidistribution results were first proven by Bohl, Sierpinski, and Weyl in 1909-1910. In Oresme's work, only vague hints in this direction can be found, as when he says that 'by means of the greatest inequality, which departs from every equality, the most just and established order is preserved'. " So didn't Oresme only notice ergodicity without proof, if that? – Lee Mosher May 16 '17 at 13:16
• Oresme did not "notice ergodicity". He had absolutely no means to STATE it. What he stated is the density of the orbits. Without proof. Density of trajectories is nowadays an exercise for a good high school student. Ergodicity is not! – Alexandre Eremenko May 16 '17 at 18:41
• I agree with Alexandre. The idea of ergodicity, as a refinement of dense orbits, is so advanced I don't understand how anyone could claim it is in any way attributable to someone from the 1300s. – KConrad May 17 '17 at 4:11

The idea to study what we call irrational rotation of a torus indeed belongs to Nicole Oresme, at least he clearly understood the density of the trajectories (which is not the same as ergodicity or equidistribution! One thing is to say that a trajectory visits every interval infinitely many times, and another thing to say HOW OFTEN. Oresme has absolutely no adequate language or tools to address, or even to state the second question). He uses the words such as "probability" too, for example he tries to say that a random number is almost surely irrational. But he really has no clear concept of a real number. With a hindsight one can read in his paper more than he really wrote. The first rigorous proof, according to the modern standards belongs to Hermann Weyl, and on my opinion it is an enormous overstatement to credit ergodicity to Oresme.

References: These two papers contain the condensed English translation with extended comments of Oresme two texts on irrational rotation:

Edward Grant, Oresme and His De Proportionibus Proportionum, Isis, Vol. 51, No. 3 (Sep., 1960), pp. 293-314, doi: 10.1086/348912, jstor.

Edward Grant, Nicole Oresme and the Commensurability or Incommensurability of the Celestial Motions, Archive for History of Exact Sciences Vol. 1, No. 4 (26.10.1961), pp. 420-458, doi: 10.1007/BF00328576, jstor.

Unlike the papers of Oresme, Weyl's paper is only available in German and Russian: Ueber die Gleichverteilung von Zahlen mod. Eins,". Math. Ann. 77 (3): 313–352, doi: 10.1007/BF01475864, eudml.

Вейль Г, Избранные труды. Математика. Теоретическая физика, М.: Наука, 1984

By the way, Oresme's main goal was to refute astrology by showing that celestial phenomena, like conjunctions and oppositions are essentially random:-) Because the periods are incommensurable. Because two random numbers are incommensurable. Exciting reading! Written around 1360. And one of the main goals of Weyl was to prove the existence of Mean Motion (conjectured by Lagrange), another problem coming from astronomy. Weyl's proof of Lagrange's conjecture contained a gap which was filled by Tornehave and Jessen in 1943. (I added this because these remarkable results seems to be almost forgotten).

• Thanks! Is this the right reference? [17] H. Weyl, Uber die Gleichverteilung von Zahlen mod. Eins, ¨ Math. Ann., 77 (1916), 313–352. – Amir Sagiv May 16 '17 at 15:33
• Yes, it is. Though it is possible Weyl had several papers on the subject. – Alexandre Eremenko May 16 '17 at 18:38

Weyl had only one "big" paper on uniform distribution theory, which is the 1916 paper. However, the fact that $(n \alpha)$ is equidistributed for irrational $\alpha$ is usually attributed to Bohl, Spierpinski and Weyl (independently), who proved it in 1909-1910. However, they did not prove anything about ergodicity. The ergodicity question does not ad-hoc have any close relation to the equidistribution problem.

• Thanks! So who proved ergodicity? – Amir Sagiv Sep 27 '17 at 13:06
• Ooops - I wanted to write this as a comment, not as a full answer. Anyway, an additional remark: there is a very interesting paper on the early history of uniform distribution theory by Jörn Steuding: mathematik.uni-wuerzburg.de/~steuding/steuding.pdf (By as to the question "who first proved ergodicity", I don't know.) – Kurisuto Asutora Sep 27 '17 at 15:54