Full deautonomisation by singularity confinement as an integrability test

Alexander Stokes, Takafumi Mase, Ralph Willox, Basile Grammaticos
Nonlinear Sciences, Exactly Solvable and Integrable Systems, Exactly Solvable and Integrable Systems (nlin.SI), Mathematical Physics (math-ph), Algebraic Geometry (math.AG), Dynamical Systems (math.DS)
2023-06-01 16:00:00
Since its introduction, the method of full deautonomisation by singularity confinement has proved a strikingly effective way of detecting the dynamical degrees of birational mappings of the plane. This method is based on an observed link between two a priori unrelated notions: firstly the dynamical degree of the mapping and secondly the evolution of parameters required for its singularity structure to remain unchanged under a sufficiently general deautonomisation. We give a proof of this conjectured correspondence for a large class of birational mappings of the plane via the spaces of initial conditions for their deautonomised versions. We show that even for non-integrable mappings in this class, the surfaces forming these spaces have effective anticanonical divisors and one can define a kind of period map similar to that in the theory of rational surfaces associated with discrete Painlev\'e equations. This provides a bridge between the evolution of coefficients in the deautonomised mapping and the induced dynamics on the Picard lattice which encode the dynamical degree.
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