162
J. Benson and F. Valiquette
Substituting these expressions into the recurrence relations for the normalized
invariants I n = ι n (x n−1 ), J n = ι n (y n−1 ), K n = ι n (x n+1 ) introduced in (3), we
obtain
dI n = ω n−1
n
− ω n
n +
2 J n
K n
(σ n+1
n
− σ n
n ) + 1
(( 2 J 2
n − I 2
n )ω n
n + 2 J n (K n − 2I n )σ n
n
,
dJ n = σ n−1
n
− σ n
n +
I n
K n
(σ n
n − σ n+1
n
) + 1
− 2I n J n ω n
n + (I 2
n − 2 J 2
n − I n K n )σ n
n
,
dK n = ω n+1
n
− ω n
n − 1 K 2
n ω n
n .
4.2 Shift Map
Let
m n = ρ n+1 ρ
−1
n =
a n b n
− 1 b n a n
,
a n a n + 1 b n b n = 1,
denote the Maurer–Cartan invariant matrix, which is an element of the Cayley–Klein
group SO 1 ,, 2 (3). Using the recurrence relations
I n+1 + i 2 J n+1 = ι n+1 (z n ) = m n · ι n (z n ) = m n · 0 =
b n
a n
,
0 = ι n+1 (z n+1 ) = m n · ι n (z n+1 ) = a n K n + b n ,
we find that
a n =
e
i 2 φ n
1 + 1 K 2
n
,
b n =
K 2
n e
−i 2 φ n
1 + 1 K 2
n
,
(5)
where
T 2 (2φ n ) =
J n+1
I n+1
.
We also obtain the syzygy
K n =
I 2
n+1 + 2 J 2
n+1 .