https://doi.org/10.1007/s100510050488
Elliptic curves from finite order recursions or non-involutive permutations for discrete dynamical systems and lattice statistical mechanics*
1
Centre de Recherches sur les Très Basses Températures,
BP 166, 38042 Grenoble, France
2
LPTHE, tour 16, 4 place Jussieu,
75252 Paris Cedex, France
Received:
9
February
1998
Revised:
13
March
1998
Accepted:
17
March
1998
Published online: 15 October 1998
We study birational mappings generated by matrix inversion
and permutations of the entries of
matrices.
For q=3 we have performed a systematic
examination of all the birational mappings
associated with permutations of
matrices in order to find integrable mappings and some finite order
recursions. This exhaustive analysis gives, among
30 462 classes of mappings, 20 classes
of integrable birational mappings, 8 classes associated with
integrable recursions
and 44 classes yielding finite order
recursions. An exhaustive analysis (with
a constraint on the diagonal entries) has also been
performed for
matrices: we have found
880 new classes of mappings
associated with integrable recursions.
We have visualized the orbits of the
birational mappings corresponding
to these 880 classes.
Most correspond to elliptic
curves and very few to surfaces or higher dimensional algebraic
varieties.
All these new examples show that integrability
can actually correspond to non-involutive permutations.
The analysis of the integrable cases
specific of a particular size of the matrix and
a careful examination of
the non-involutive permutations,
shed some light on the integrability
of such birational mappings.
PACS: 05.50.+q – Lattice theory and statistics; Ising problems / 02.10.-v – Logic, set theory, and algebra / 02.20.-a – Group theory
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 1998