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We provide new examples of integrable rational maps in four dimensions with two rational invariants, which have unexpected geometric properties, as for example orbits confined to non algebraic varieties, and fall outside classes studied by earlier authors. We can reconstruct the map from both invariants. One of the invariants defines the map unambiguously, while the other invariant also defines a new map leading to non trivial fibrations of the space of initial conditions.
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We prove that under mild hypothesis rational maps on a surface preserving webs are of Latt`es type. We classify endomorphisms of P^2 preserving webs, extending former results of Dabija-Jonsson.
In this paper we present a class of four-dimensional bi-rational maps with two invariants satisfying certain constraints on degrees. We discuss the integrability properties of these maps from the point of view of degree growth and Liouville integrability.
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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