Article Dans Une Revue Journal of Physics A: Mathematical and Theoretical Année : 2025

Obtaining the growth of higher order mapping through the study of singularities

Résumé

We examine a collection of integrable and non-integrable birational mappings of order higher than two and show that the methods developed for calculating the degree growth of second order mappings can be transposed to higher-order systems. In particular, Halburd's method which allows one to obtain explicitly the degree growth of a second order mapping, based on the patterns of its singularities, will be shown to work perfectly well in the higher-order case. The only precaution one must take is to choose initial conditions that are specifically adapted to the method. In the case of non-integrable mappings, one can apply either the express variant of Halburd's method or the full-deautonomisation approach and we show that both lead to the (same) exact expression for the dynamical degree of the mapping.

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hal-04598791 , version 1 (03-06-2024)

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A. Ramani, B. Grammaticos, A.S. Carstea, R. Willox. Obtaining the growth of higher order mapping through the study of singularities. Journal of Physics A: Mathematical and Theoretical, 2025, 58 (11), pp.115201. ⟨10.1088/1751-8121/adbd0c⟩. ⟨hal-04598791⟩
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