Danesh Jogia et al 2006 J. Phys. A: Math. Gen. 39 1133 doi:10.1088/0305-4470/39/5/008
Danesh Jogia1, John A G Roberts1 and Franco Vivaldi2
Show affiliationsWe show that the dynamics of a birational map on an elliptic curve over a field is, typically, conjugate to addition by a point (under the associated group law). When the field is taken to be the function field of rational complex functions of one variable, this amounts to an algebraic geometric version of the Arnold–Liouville integrability theorem for planar integrable maps. By-products of this approach are that birational maps preserving foliations are necessarily the composition of two involutions, and that relationships between birational maps preserving the same foliation can be described in terms of the respective points they add on the corresponding Weierstrass curves. When the result is applied to finite fields, it helps explain some universal features of the periodic orbit distribution function for the reductions of integrable maps.
02.40.-k Geometry, differential geometry, and topology
Issue 5 (3 February 2006)
Received 19 September 2005, in final form 2 December 2005
Published 18 January 2006
Danesh Jogia et al 2006 J. Phys. A: Math. Gen. 39 1133
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