Renormalization of a one-parameter family of piecewise isometries

We consider a one-parameter family of piecewise isometries of a rhombus. The rotational component is fixed, and its coefficients belong to the quadratic number field $K=\mathbb{Q}(\sqrt{2})$. The translations depend on a parameter $s$ which is allowed to vary in an interval. We investigate renormalizability. We show that recursive constructions of first-return maps on a suitable sub-domain eventually produce a scaled-down replica of this domain, but with a renormalized parameter $r(s)$. The renormalization map $r$ is the second iterate of a map $f$ of the generalised L\"uroth type (a piecewise-affine version of Gauss' map). We show that exact self-similarity corresponds to the eventually periodic points of $f$, and that such parameter values are precisely the elements of the field $K$ that lie in the given interval. The renormalization process is organized by a graph. There are ten distinct renormalization scenarios corresponding to as many closed circuits in the graph. The process of induction along some of these circuits involves intermediate maps undergoing, as the parameter varies, infinitely many bifurcations. Our proofs rely on computer-assistance.