Dynamics and Chaos 2022-2023, Homework set 2
Please answer the exercises below and hand in your solutions before Tuesday March 14, 23:59.
When homework that is handed in after the deadline, you will receive feedback, but the
homework will not be graded.
Please note that you are allowed to discuss the exercises with your fellow students, but each
student should hand in their own solutions. Handing in solutions that are very similar to other
students’ solutions is seen as fraude and has to be reported to the board of examiners.
Exercise 1
Define the transformation T : [1, 7] → [1, 7] by setting

3x + 1,



9 − x,
T (x) =

13 − 2x,



8 − x,
if
if
if
if
1 ≤ x < 2,
2 ≤ x < 4,
4 ≤ x < 5,
5 ≤ x ≤ 7.
(i) Find the fixed point of T . Is it stable? Is it attracting?
(ii) Prove that T has a periodic point of period m ≥ 1 for all 5 . m, where . is the Sharkovsky ordering.
(iii) Prove that T does not have periodic points of period 3 and 5.
Exercise 2
Let T : T → T be a circle homeomorphism and F : R → R a lift of T . Prove that if T is a homeomorphism,
then F is strictly monotone.
N.B. In the lecture notes there is an exercise asking you to prove that if T is an orientation preserving
homeomorphism, then for each x ∈ R and each k ∈ Z it holds that F (x + k) = F (x) + k (there was a typo
in the exercise). If you plan to use this statement to solve Exercise 2, then please also prove this statement.
Exercise 3
Prove or disprove the following statement:
A continuous transformation T : I → I defined on an interval I ⊆ R and for which the set of periodic points
of T is dense in I has sensitive dependence on initial conditions.
1