In his famous paper on Nash-Moser Theory, Hamilton mentioned in Counterexample 5.5.2 that a rotation $f: \theta\mapsto\theta + 2\pi/k$ on a circle can be "pushed a little bit" so that the new diffeomorphism only has $0$ as a $k$-periodic point, but $f(\pi/k)\neq3\pi/k$. My question is: is there any reference where Hamilton's construction was explicitly expressed? How can one "push" the rotation to obtain a diffeomorphism that only has $0$ as its $k$-periodic point?
Thank you in advance!