3.5 Renormalization in Twist Maps
67
to the same M i -cycle is x i = ±
1
F i
. More generally, for a fixed point with ω i =
F i−1
F i
, p i = ω i+1 − ω i and x i = (−1) i 1
F i
.
Let us now move the origin of coordinates (p, x) to the position of the dominant
elliptic fixed point with winding number ω i . Then its two nearest neighbors lie at
(p = ω i+1 −ω i , x = 0) and (p = 0, x = (−1) i 1
F i
). We next rescale the coordinates
so that the nearest neighbors lie at (p = 1, x = 0) and (p = 0, x = 1). This can be
done by magnifying the neighborhood of the fixed point at the origin by a sequence
of mappings,
1
1
= B 0 B 1 . . . B i−1
ω i+1 − ω i
(−1) i 1
F i
,
(3.35)
where
B i−1 =
−
W i−1
W i
0
0
−
F i
F i−1
(3.36)
and W i = ω i − ω i+1 . With this scale change, the neighborhood of each dominant
elliptic fixed point gets mapped onto the unit square. Note that in the limit i → ∞,
B i−1 →
−γ 2 0
0 −γ
,
(3.37)
and the scale change becomes independent of i.
The mapping that reproduces the neighborhood of the ith elliptic fixed point on
the unit square is
G
R
= B 0 . . . B i−1 G
M i R
N i B
−1
i−1 . . . B
−1
0 .
(3.38)
Let us now consider one step in the renormalization transformation from G M i R N I
to G M i−1 R N i−1 ,
G
M i−1 R
N i−1 = B i−1 G
M i R
N i B
−1
i−1 .
(3.39)
If we use the relation F i+1 = F i−1 + F i for the Fibonacci numbers, we can write
Eq. (3.39) in the form
G
M i−1 R
N i−1 = B i−1 G
M i−1 R
N i−1 G
M i−2 R
N i−2 B
−1
i−1 .
(3.40)
We now define T i−1 = G M i−1 R N i−1 and U i−1 = G M i−2 R N i−2 = B i−2 T i−1 B
−1
i−2 .
Then Eq. (3.40) can be written in the form
T i−1 = B i−1 T i−1 U i−1 B
−1
i−1
(3.41)
67
to the same M i -cycle is x i = ±
1
F i
. More generally, for a fixed point with ω i =
F i−1
F i
, p i = ω i+1 − ω i and x i = (−1) i 1
F i
.
Let us now move the origin of coordinates (p, x) to the position of the dominant
elliptic fixed point with winding number ω i . Then its two nearest neighbors lie at
(p = ω i+1 −ω i , x = 0) and (p = 0, x = (−1) i 1
F i
). We next rescale the coordinates
so that the nearest neighbors lie at (p = 1, x = 0) and (p = 0, x = 1). This can be
done by magnifying the neighborhood of the fixed point at the origin by a sequence
of mappings,
1
1
= B 0 B 1 . . . B i−1
ω i+1 − ω i
(−1) i 1
F i
,
(3.35)
where
B i−1 =
−
W i−1
W i
0
0
−
F i
F i−1
(3.36)
and W i = ω i − ω i+1 . With this scale change, the neighborhood of each dominant
elliptic fixed point gets mapped onto the unit square. Note that in the limit i → ∞,
B i−1 →
−γ 2 0
0 −γ
,
(3.37)
and the scale change becomes independent of i.
The mapping that reproduces the neighborhood of the ith elliptic fixed point on
the unit square is
G
R
= B 0 . . . B i−1 G
M i R
N i B
−1
i−1 . . . B
−1
0 .
(3.38)
Let us now consider one step in the renormalization transformation from G M i R N I
to G M i−1 R N i−1 ,
G
M i−1 R
N i−1 = B i−1 G
M i R
N i B
−1
i−1 .
(3.39)
If we use the relation F i+1 = F i−1 + F i for the Fibonacci numbers, we can write
Eq. (3.39) in the form
G
M i−1 R
N i−1 = B i−1 G
M i−1 R
N i−1 G
M i−2 R
N i−2 B
−1
i−1 .
(3.40)
We now define T i−1 = G M i−1 R N i−1 and U i−1 = G M i−2 R N i−2 = B i−2 T i−1 B
−1
i−2 .
Then Eq. (3.40) can be written in the form
T i−1 = B i−1 T i−1 U i−1 B
−1
i−1
(3.41)