complex plot with input just from the unit circle

[ecurry]

Background: I have a function that outputs complex values.  I only care about input from the unit circle in the complex plane: (r,t) with r = |z| = 1 and t = arg z in [0,1] (or [0,2*pi] if you prefer).  I could just make a 3D plot with input the single real variable t and output x = Re f(t), y = Im f(t), except that the function I want to plot is f^n(t) - a self-composition, where I want to investigate what happens as n -> infinity.  So this would require re-writing my original function f(t) as a function f(x,y) even though it's actually a polar function f(r,t) with r=1.  Which I could do... but this work-around is now starting to feel really clunky.

Q1: Is there a better way?

Q2: Alternatively, is there an easy (less clunky that my alternative work-around) way to superimpose a (black or white) unit circle on top of my complex_plot of f^n?

Thanks!

[ecurry]

(Or, two circles, one on each side of the unit circle, so that I can still see the values I want, but have an easier time picking them out of the whole plot.)