Abstract
We introduce non trivial two‐dimensional and three‐dimensional mappings. These mappings are birational transformations, the iterates of which give a nonlinear representation of infinite Coxeter groups. These mappings originate from integrable models in statistical mechanics. They exhibit a number of remarkable properties. Some have algebraic invariants allowing for orbits lying on some smooth manifolds. Others give examples of mappings the iterates of which are Hofstadte‐like patterns and exemplify onset of order from disorder (Saturn’s ring).
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