Renormalization of one-parameter families of piecewise isometries

We consider two one-parameter families of piecewise isometries of a rhombus. The rotational component is fixed, and its coefficients belong to a quadratic number field . The translations depend on a parameter <i>s</i> which is allowed to vary in an interval. We investigate renormalizability, and show that recursive constructions of first-return maps on a suitable subdomain eventually produce a scaled-down replica of this domain, but with a renormalized parameter <i>r</i>(<i>s</i>). We treat two quadratic fields: <i>d</i> = 5, 2. In the first case, the renormalization map <i>r</i> is of Lüroth type (a piecewise-affine version of Gauss’ map), whereas in the second case, it is the second iterate of a map <i>f</i> of this type. We show that exact self-similarity corresponds to the eventually periodic points of <i>r</i> (resp. <i>f</i>), and that such parameter values are precisely the elements of the quadratic field that lie in the given interval. The renormalizability proof for is based on a straightforward application of return-map analysis. The octagonal case is far more challenging. The proof is organized by a graph analogous to those used to construct renormalizable interval-exchange transformations. There are 10 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.

我们研究两类单参数族的菱形分段等距映射。其旋转分量固定不变，且该分量的系数属于某一二次数域（quadratic number field）；平移分量则依赖于可在某区间内变化的参数s。我们针对其可重整性展开研究，并证明：在合适的子区域上构造首次回归映射的递归过程，最终将得到原区域的一个缩放复制品，且其对应的重整化参数为r(s)。

我们分别针对两类二次数域展开研究，其判别式分别为5与2。在第一种情形下，重整化映射r属于吕罗特（Lüroth）型映射，即高斯映射的分段仿射版本；而在第二种情形下，该映射则是此类吕罗特型映射f的二次迭代。我们证明，精确自相似性对应于映射r（对应第一种情形）或f（对应第二种情形）的最终周期点，且此类参数值恰好是落在给定区间内的二次数域元素。

针对第一类菱形分段等距映射的可重整性证明，可通过直接应用回归映射分析完成。而八边形分段等距映射的情形则要复杂得多。该证明通过类比用于构造可重整区间交换变换（interval-exchange transformations）的图结构来组织，图中存在10种不同的重整化场景，分别对应10个闭合回路。在部分回路中进行归纳推导时，随着参数的变化，中间映射会出现无穷多的分岔现象。我们的全部证明均借助计算机辅助完成。