An Elementary Construction of Modified Hamiltonians and Modified Measures of 2D Kahan Maps

We show how to construct in an elementary way the invariant of the KHK discretisation of a cubic Hamiltonian system in two dimensions. That is, we show that this invariant is expressible as the product of the ratios of affine polynomials defining the prolongation of the three parallel sides of a hexagon. On the vertices of such a hexagon lie the indeterminacy points of the KHK map. This result is obtained analysing the structure of the singular fibres of the known invariant. We apply this construction to several examples, and we prove that a similar result holds true for a case outside the hypotheses of the main theorem, leading us to conjecture that further extensions are possible.