C2 F
T
D
S
E
S
3
Stars indicate difficulty of questions.
1.* Let f be a continuous map. Show that the
ω
-limit set
ω
( x ) is i) invariant ii) closed
2.* Let f : [0,1]
×
[0,1]
→
[0,1]
×
[0,1] be the thin Baker’s transformation on the unit square: f ( x , y ) =


 (
( 2
2 x x
,
−
1
3
1 y
,
)
1
3 y
+
2
3
)
0
1
2
1
2
1
Find periodic orbits of (minimal) periods 2, 3 and 4 respectively. Hint: it may help to use the binary and base 3 expansions described in lectures.
3.*** Let g : [0,1]
→
[0,1] be the doubling map g ( x ) = 2 x mod 1 described in lectures. Can you find a minimal set which is not a periodic orbit? Hence, or otherwise, do the same for the Baker’s transformation in Q2.
4.
Let f be the rigid rotation rotation of the circle f(x) = x +
ω
(mod 1). Show that
ω( x) is a minimal set for every x . Hint: it may help to consider separately the cases where
ω is rational/irrational.
5.* Suppose that f is a contraction mapping on a complete metric space. Show that the unique fixed point is asymptotically stable.
6.
Discuss the stability/asymptotic stability of periodic orbits in the Baker’s transformation in Q2, in the map g ( x ) = 2 x (mod 1) ( e.g.
Q3 above) and in the map f (
θ
) =
θ
+
ω
(mod 1) for rational
ω
( e.g.
Q4 above).
7.
Consider the simple harmonic oscillator
..
x +
α
2 x = 0
Write this as a linear differential equation in two dimensions and discuss the stability of the origin. Show that apart from the origin, every point lies on a periodic orbit. Compute the period and discuss the stability of these periodic orbits.
8.
Find co-ordinates u and v (as functions of x and
.
x ) such that the oscillator in Q7 can be written as du dt dv dt
=
=
-
α
α u v
Hence compare your answers to Q7 with Q12 on Example Sheet 1.

C2 Exercise Sheet 3 2
9.
Suppose that f : X
→
X and g : Y
→
Y are conjugate ( i.e.
there exists an invertible map h : Y
→
X such that f ° h = h ° g ). Show that if y is a periodic orbit of g then h ( y ) is a periodic orbit of f with the same period. This is the converse of what was shown in lectures.
10. As Q9, but now show that the minimal period of y and h ( y ) are the same.
11. Suppose that h is a topological conjugacy between f : X
→
X and g : Y
→
Y , (so that h and h -1 are continuous). Show that if y is a stable/a. stable periodic orbit of g , then h ( y ) is a stable/a. stable periodic orbit of f . This is the converse of what was shown in lectures.
12. Suppose that h is a conjugacy between f : X
→
X and g : Y
→
Y . Show that f n ° h is also a conjugacy, for any n .
13. Suppose that h is a topological conjugacy between f : X
→
X and g : Y
→
Y . Show that if A is a minimal invariant set of g , then h ( A ) is a minimal invariant set of f , and vice versa .
14. Suppose that h : Y
→
X is a semi-conjugacy between f : X
→
X and g : Y
→
Y , so that f ° h = h ° g , h ( Y )
= X , but h is not necessarily invertible. Show that if y is a periodic orbit of g then h ( y ) is a periodic orbit of f with the same period. Are the minimal periods necessarily the same?
15. Suppose that h : Y
→
X is a continuous semi-conjugacy between f : X
→
X and g : Y
→
Y and the orbit of y is dense. Show that the orbit of h ( y ) is dense.
16. Let
Γ
be a periodic orbit of period q of a map f : R n
→ R n . Show that the eigenvalues of D independent of the choice of point x
∈Γ
.
x f q are
17. Let
Γ
be a periodic orbit of a differential equation on R n . Let
Σ
0
and
Σ
1 and g
0
, g
1
the corresponding return maps. Let x fixed points of g
0
and g
1
0
∈Σ
0
∩ Γ
and x
1
∈Σ
1
respectively. Show that the eigenvalues of D
∩ Γ
be the corresponding x
0
be two Poincaré sections g
0
and D
1
are the same.
Hint: show that g
0
and g
1
are smoothly conjugate.
x
1 g
18.** Show that f : R → R and g : R → R are conjugate in a neighbourhood of 0, where f ( x ) =
λ
for some 0 <
λ
< 1 and g ( x ) = x - x
3. Hint: show all nearby points tend to 0 under both maps, pick an arbitrary x > 0, define h linearly on the interval [ g ( x ), x ] and extend using f ° h = h ° g , and show that the resulting h is continuous; repeat for x < 0.
19. C2 May 1994, Q3.
20. C2 May 1994, Q4.
21. C2 May 1995, Q3.
22. C2 May 1994, Q5.
23. C2 May 1995, Q5i).
24. C2 May 1995, Q5ii).