What, if any, is the relation between Cantor's function and Ligeti studio: Devil's Staircase?
There is a Masters thesis by Lauren Halsey entitled, "An examination of rhythmic practices and influences in the keyboard works of György Ligeti" (UNCG link), which addresses your question:
"...the idea for this etude emerged: “an endless climbing, a wild apocalyptic vortex, a staircase it was almost impossible to ascend.”28 This etude shares the name and characteristics of the mathematical concept of a “devil’s staircase.” This phenomenon, based on Cantor Sets, involves the relationship of disproportional segments combining to create a self-similar group.29 This concept is also used in the “mode locking” features of clocks and pendulums.30 Ligeti expresses this concept with the inclusion of groups of two and three eighth notes that, when combined, create a self-similar rhythmic set. This grouping structure creates pulse streams and defines the formal boundaries of this piece. The structures seem to spiral infinitely up the piano, suddenly falling down to the lowest octaves.31
And here is a graph from Wikipedia's Cantor function article of the Devil's Stairase: