4.2 The Complete Ternary Horseshoe
109
Fig. 4.6 The outgoing
momentum p out versus phase
angles χ of the incident
trajectories (plot by Christof
Jung)
4.2.2.2 Symbolic Dynamics
We can assign a symbol sequence to the increasingly complex stretching and folding
pattern of the fundamental region under the action of the map. In Fig. 4.7, we assign
symbols A, B, and C to the regions where mapped sections of the fundamental area
cross the boundaries of the fundamental area. If we compare Figs. 4.3b and 4.7a,
which show the first iteration of the map, we see how the symbol pattern is assigned.
If we compare Figs. 4.3c and 4.7b, the pattern of crossings is the same, but the
symbol sequence is more complex and becomes harder to follow with increasing
number of iterations of the map. In Fig. 4.7c, we show the symbol structure of three
iterations of the map. The complexity of the symbol structure grows exponentially,
although there is a very simple pattern underlying it.
In Fig. 4.8, we show a symbol tree that more clearly shows the evolution of the
pattern of tendrils crossing the boundary of the fundamental region. At each level,
the pattern of symbols in central clusters of symbols is reversed from that of the
symbol sequence of the lower level of the map. This symbol tree clearly shows
that scattering from the complete ternary horseshoe forms is a self-similar fractal
process.
4.2.2.3 Incomplete Ternary Horseshoes
In the previous section, we examined a scattering system whose dynamics, in
certain parameter ranges, was described by a complete ternary horseshoe. However,
dynamics of most scattering systems is governed by incomplete binary or ternary
horseshoes. Many examples of scattering systems with incomplete horseshoes have
been studied in the literature. Measures for determining the degree of “incompleteness” of the horseshoe development have been discussed by Jung et al. (1999). In
subsequent sections, we will see how this horseshoe development can manifest itself
in some atomic and molecular systems.
109
Fig. 4.6 The outgoing
momentum p out versus phase
angles χ of the incident
trajectories (plot by Christof
Jung)
4.2.2.2 Symbolic Dynamics
We can assign a symbol sequence to the increasingly complex stretching and folding
pattern of the fundamental region under the action of the map. In Fig. 4.7, we assign
symbols A, B, and C to the regions where mapped sections of the fundamental area
cross the boundaries of the fundamental area. If we compare Figs. 4.3b and 4.7a,
which show the first iteration of the map, we see how the symbol pattern is assigned.
If we compare Figs. 4.3c and 4.7b, the pattern of crossings is the same, but the
symbol sequence is more complex and becomes harder to follow with increasing
number of iterations of the map. In Fig. 4.7c, we show the symbol structure of three
iterations of the map. The complexity of the symbol structure grows exponentially,
although there is a very simple pattern underlying it.
In Fig. 4.8, we show a symbol tree that more clearly shows the evolution of the
pattern of tendrils crossing the boundary of the fundamental region. At each level,
the pattern of symbols in central clusters of symbols is reversed from that of the
symbol sequence of the lower level of the map. This symbol tree clearly shows
that scattering from the complete ternary horseshoe forms is a self-similar fractal
process.
4.2.2.3 Incomplete Ternary Horseshoes
In the previous section, we examined a scattering system whose dynamics, in
certain parameter ranges, was described by a complete ternary horseshoe. However,
dynamics of most scattering systems is governed by incomplete binary or ternary
horseshoes. Many examples of scattering systems with incomplete horseshoes have
been studied in the literature. Measures for determining the degree of “incompleteness” of the horseshoe development have been discussed by Jung et al. (1999). In
subsequent sections, we will see how this horseshoe development can manifest itself
in some atomic and molecular systems.