Complexity and integrability in 4D bi-rational maps with two invariants

TL;DR

This paper presents fourth-order autonomous recurrence relations with two invariants exhibiting cubic or exponential degree growth, challenging the belief that maps with many invariants are limited to quadratic growth and exploring their integrability properties.

Contribution

It introduces new examples of maps with multiple invariants showing higher-than-expected degree growth, expanding understanding of integrability in 4D bi-rational maps.

Findings

Examples with cubic and exponential growth contradict previous assumptions

Cubic growth may be linked to non-elliptic fibrations of invariants

Exponentially growing cases may lack conditions for discrete Liouville theorem

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.