Full deautonomisation by singularity confinement as an integrability test

Author:

Alexander Stokes, Takafumi Mase, Ralph Willox, Basile Grammaticos

Keyword:

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)

journal:

date:

2023-06-01 16:00:00

Abstract

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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