Maximal growth and coupling structure for integrable mappings on C^N

Prove that for integrable birational mappings on C^N the maximal possible degree growth is d_n ∼ n^N and demonstrate that this maximal growth can only be realized by coupling at least one second-order mapping that has quadratic degree growth with a sufficient number of linear or linearizable mappings.

Background

Building on results for Gambier-type constructions and bounds arising from reductions of discrete KP-type equations, the authors synthesize known facts to propose a structural characterization of how maximal polynomial growth might be achieved in integrable higher-order mappings.

They conjecture that the extremal growth rate n^N for integrable maps requires coupling a quadratic-growth second-order component with additional linear or linearizable components. Establishing this would clarify the mechanism by which maximal growth arises in integrable higher-order systems.