Concerning (b), you may find this post helpful. It's not exactly the same problem, so feel free to post your thoughts.
Question:
For some m = p/q in the rational numbers consider the map f_m = [0,1) -> [0,1) defined by f_m(x) = (x + m) mod 1 for all x element of [0,1).
a) Show that f(superscript q, subscript m) = Identity on [0,1)
(my comment): This refers to the composition of f_m, q times, where q is the denominator of the rational number.
b) Now suppose we consider m in the irrationals, what happens to the iterates f(superscript n, subscript m)(x) for any point x as n gets larger?
Attempt:
For (a) I have done several "examples" to verify to myself this works. However I am not sure how to go about constructing a proof. My main problem stems from the uncertainty in representing f(superscript q, subscript m).
I know it is just (f_m o f_m o f_m ... o f_m) q times, but how do you generalize something like that?
(b) I'm not sure on how to proceed. Does it also happen to approach the identity? (just a guess)
Edit: I got (a) just need help on (b).