32
C. M. Viallet
Fig. 6 Localisation of the factors Ω
4.4 What About Entropy?
The factors Ω contain all the information on
– How the successive iterates hit singularities (remember the picture in the plane,
but now we work in an infinite dimensional space of initial conditions)
– the sequence of degrees of the iterates and leading to the vanishing of the entropy.
The sequence of degrees for this specific model is obtained by direct calculation
{d n } = 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, . . .
(20)
It can be fitted by the generating function
g(s) =
∞
k=0
d k s
k
=
1 − s + s 2
(1 − s) 3
(21)
that is to say
d n = 1 +
n(n + 1)
2
(22)
quadratic growth, vanishing entropy.
Remark the function g can be guessed from the first 5 iterations, and the further
terms of the sequence give a check. Moreover the result can be proved along the
lines of [41].
C. M. Viallet
Fig. 6 Localisation of the factors Ω
4.4 What About Entropy?
The factors Ω contain all the information on
– How the successive iterates hit singularities (remember the picture in the plane,
but now we work in an infinite dimensional space of initial conditions)
– the sequence of degrees of the iterates and leading to the vanishing of the entropy.
The sequence of degrees for this specific model is obtained by direct calculation
{d n } = 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, . . .
(20)
It can be fitted by the generating function
g(s) =
∞
k=0
d k s
k
=
1 − s + s 2
(1 − s) 3
(21)
that is to say
d n = 1 +
n(n + 1)
2
(22)
quadratic growth, vanishing entropy.
Remark the function g can be guessed from the first 5 iterations, and the further
terms of the sequence give a check. Moreover the result can be proved along the
lines of [41].