lets say we have f(x)=x+a on the circle such that a is irrational and there is an Arc of length epsilon which f^q(a) never enters for any q. then f^n(a) gets you back to the starting point otherwise x+qa mod 1 would eventually get you inside the arc of length epsilon. hence f^n(a)=a so that x+na mod 1 = x which implies that a is in Q. but a by definition is irrational