QuestionQuestion

6. Describe the phase portrait of the map of the circle given by f(θ) = θ + 2π/n + ϵ sin(n θ)
for 0 < ϵ < 1/n.

7. Prove that a homeomorphism of R can have no periodic points with prime period greater than 2. Give an example of a homeomorphism that has a periodic point of period 2.

8. Prove that a homeomorphism cannot have eventually periodic points.

9. Let S: S¹ --> S¹ be given by S(θ) = θ + ω + ϵ sin(θ) where ω and ϵ are constants. Prove that S is a homeomorphism of the circle if |ϵ| < 1.

10. Let f(θ) = 2θ be the map of S¹. Prove that periodic points of f are dense in S¹.

11. Prove that eventually fixed points for the map are also dense in S¹.

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