84
3 Area-Preserving Maps
Fig. 3.23 The flux, , associated with the periodic orbits with winding number ω =
[a 0 , . . . , a m+1 ] = [a 0 , . . . , a m , 1] (values of the entries a i are indicated on the figure) plotted as
a function of X, the horizontal coordinate in the universal map, when the golden mean is critical.
Note that the flux associated with the rational approximates to the golden mean tends to zero as
m → ∞, whereas the flux associated with rational approximates for other KAM tori tends to finite
values (indicating that they are actually cantori) (based on data from (MacKay 1982))
W
δ
=
1
αβ
W ((K).
(3.69)
From Eq. (3.69), we can find the dependence of W on K. Let ((K) =
A((K) η , where A is a constant and η is an exponent to be determined. Then
Eq. (3.69) implies that δ η = αβ or
η = ln δ (αβ) = 3.011722.
(3.70)
This scaling behavior of the flux associated with the rational approximates has been
used to develop a theory of diffusion of trajectories, in mixed phase spaces, in terms
of a self-similar Markov tree.
3.8 Renormalization Map
The behavior of the region of phase space between any two neighboring primary
resonances is largely determined by those two resonances (as long as resonances
outside the region have not overlapped with them). The effects of primary reso-
3 Area-Preserving Maps
Fig. 3.23 The flux, , associated with the periodic orbits with winding number ω =
[a 0 , . . . , a m+1 ] = [a 0 , . . . , a m , 1] (values of the entries a i are indicated on the figure) plotted as
a function of X, the horizontal coordinate in the universal map, when the golden mean is critical.
Note that the flux associated with the rational approximates to the golden mean tends to zero as
m → ∞, whereas the flux associated with rational approximates for other KAM tori tends to finite
values (indicating that they are actually cantori) (based on data from (MacKay 1982))
W
δ
=
1
αβ
W ((K).
(3.69)
From Eq. (3.69), we can find the dependence of W on K. Let ((K) =
A((K) η , where A is a constant and η is an exponent to be determined. Then
Eq. (3.69) implies that δ η = αβ or
η = ln δ (αβ) = 3.011722.
(3.70)
This scaling behavior of the flux associated with the rational approximates has been
used to develop a theory of diffusion of trajectories, in mixed phase spaces, in terms
of a self-similar Markov tree.
3.8 Renormalization Map
The behavior of the region of phase space between any two neighboring primary
resonances is largely determined by those two resonances (as long as resonances
outside the region have not overlapped with them). The effects of primary reso-