26
C. M. Viallet
Fig. 2 A symmetry pattern
Σ
Σ’
Q
P
Fig. 3 A singularity pattern in two dimensions
3.5 Singularity Analysis
There is a deep link between integrability and singularity structure. This structure is
best illustrated in the case of maps, that is to say ordinary difference equations.
Figure 3 shows what a birational map in two dimensions can do. The curve Σ
is sent to a point P by the evolution. This means that the backward evolution is
not defined at P . The point P is then sent to the point Q, which is singular as the
backward evolution sends the whole curve Σ to Q.
C. M. Viallet
Fig. 2 A symmetry pattern
Σ
Σ’
Q
P
Fig. 3 A singularity pattern in two dimensions
3.5 Singularity Analysis
There is a deep link between integrability and singularity structure. This structure is
best illustrated in the case of maps, that is to say ordinary difference equations.
Figure 3 shows what a birational map in two dimensions can do. The curve Σ
is sent to a point P by the evolution. This means that the backward evolution is
not defined at P . The point P is then sent to the point Q, which is singular as the
backward evolution sends the whole curve Σ to Q.