72
G. Gubbiotti
Applying the method presented in Remark 1 we find that the map (13) admits the
following functionally independent invariants:
I 1 =
x n x n−2
x n−1
+
x n−1 x n+1
x n
+ x n + x n−1 + 2x n x n−2
+ 2x n−1 x n+1 + 2x n x n−1 + x n x n−1 x n−2 + x n x n−1 x n+1
(16a)
I 2 =
x n−2
x n−1
2
+
x n+1
x n
2
+ x
2
n−1 x
2
n−2 + x
2
n x
2
n+1 + x
2
n x
2
n−1
− 2
x n−2 x n+1
x n
x n−1
+
1
x n x n−1
+
x n−1
x n
+ x n x n−1
+ x n−1 x n (x n−1 x n−2 + x n x n+1 )
+ 4
x n−1 x
2
n−2 + x n x
2
n+1 + (x n + x n−1 ) (1 − x n−2 x n+1 )
+ 4
x n+1 (x n−1 + x n+1 − x n−2 )
x n
+
x n−2 (x n + x n−2 − x n+1 )
x n−1
+ 6
x
2
n+1 + x n−1 x n+1 + x n x n−2 + x
2
n−2
+ 8 (x n x n−1 − x n−2 x n+1 )
(16b)
I 3 =
N 3
D 3,1 D 3,2
,
(16c)
where
N 3 = x
3
n x
2
n−2 x
3
n−1 + x
3
n x n−2 x
3
n−1 x n+1 + x
3
n x
3
n−1 x
2
n+1
+ 3x
3
n x
2
n−2 x
2
n−1 + 3x
3
n x n−2 x
3
n−1 + 3x
3
n x
3
n−1 x n+1
+ 3x
3
n x
2
n−2 x n−1 + 4x
3
n x n−2 x
2
n−1 − x
3
n x n−2 x n−1 x n+1
(17a)
+ 2x
3
n x
3
n−1 + x
3
n x
2
n−1 x n+1 + x
2
n x n−2 x
3
n−1 − x
2
n x n−2 x
2
n−1 x n+1
+ 4x
2
n x
3
n−1 x n+1 − x n x n−2 x
3
n−1 x n+1 + 3x n x
3
n−1 x
2
n+1 + x
3
n x
2
n−2
+ x
3
n x n−2 x n−1 + x
3
n x
2
n−1 + x n
2 x n−2 x
2
n−1 − x
2
n x n−2 x n−1 x n+1
+ x
2
n x
2
n−1 x n+1 − x n x n−2 x n−1
2 x n+1 + x n x
3
n−1 x n+1 + x
3
n−1 x
2
n+1
+ x
2
n x
2
n−1 + 3x
2
n x
3
n−1 x
2
n+1 + x
2
n x
3
n−1 ,
D 3,1 = x n x
2
n−1 x n−2 + 2x n x n−1 x n−2 + x n x
2
n−1
− x n x n−1 x n+1 + x n x n−2 − x n−1 x n+1 ,
(17b)
D 3,2 = x n x n−1 x n−2 − x
2
n x n−1 x n+1 − x
2
n x n−1
− 2x n x n−1 x n+1 + x n x n−2 − x n−1 x n+1 .
(17c)