In this letter we give fourth-order autonomous recurrence relations with two invariants, whose degree growth is cubic or exponential. These examples contradict the common belief that maps with sufficiently many invariants can have at most quadratic growth. Cubic growth may reflect the existence of non-elliptic fibrations of invariants, whereas we conjecture that the exponentially growing cases lack the necessary conditions for the applicability of the discrete Liouville theorem.
Complexity and Integrability in 4D Bi-rational Maps with Two Invariants / G. Gubbiotti, N. Joshi, D.T. Tran, C. Viallet (SPRINGER PROCEEDINGS IN MATHEMATICS & STATISTICS). - In: Asymptotic, Algebraic and Geometric Aspects of Integrable Systems : In Honor of Nalini Joshi On Her 60th Birthday / [a cura di] F. Nijhoff, Y. Shi, Da-jun Zhang. - Prima edizione. - [s.l] : Springer Nature, 2020. - ISBN 978-3-030-56999-0. - pp. 17-36 (( convegno TSIMF tenutosi a Sanya nel 2018 [10.1007/978-3-030-57000-2_2].
Complexity and Integrability in 4D Bi-rational Maps with Two Invariants
G. Gubbiotti
Primo
;
2020
Abstract
In this letter we give fourth-order autonomous recurrence relations with two invariants, whose degree growth is cubic or exponential. These examples contradict the common belief that maps with sufficiently many invariants can have at most quadratic growth. Cubic growth may reflect the existence of non-elliptic fibrations of invariants, whereas we conjecture that the exponentially growing cases lack the necessary conditions for the applicability of the discrete Liouville theorem.
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