Abstract
We give a criterion to determine when the degree growth of a birational map of the complex projective plane which fixes (the action on the basis of the fibration is trivial) a rational fibration is linear up to conjugacy. We also compute the centraliser of such maps. It allows us to describe the centraliser of the birational maps of the complex projective plane which preserve a rational fibration (the action on the basis of the fibration being not necessarily trivial); this question is related to some classical problems of difference equations.
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