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).

rationalvalues of $\alpha$ should be expected to lead to ergodic rotations, no? $\endgroup$ – Kaz May 17 '17 at 17:45