Deformed cluster maps of type A₂₍

We extend recent work of the third author and Kouloukas by constructing deformations of integrable cluster maps corresponding to the Dynkin types A₂₍, lifting these to higher-dimensional maps possessing the Laurent property and demonstrating integrality of the deformations for N 3. This provides the first infinite class of examples (in arbitrarily high rank) of such maps and gives information on the associated discrete integrable systems. Key to our approach is a ``local expansion'' operation on quivers which allows us to construct and study mutations in type A₂₍ from those in type A₂ (₍-₁).