Continuous self-similarity in parametric piecewise isometries

TL;DR

This paper explores two types of continuous self-similarity in parametric piecewise isometries of a rhombus, revealing conditions under which these systems exhibit self-similarity based on algebraic parameters and phase space structure.

Contribution

It introduces two renormalization scenarios for PWI of a rhombus, demonstrating how algebraic parameters influence self-similarity and phase space invariants, with exact algebraic computations.

Findings

Self-similarity occurs when one parameter is in  and the other is free.

Phase space splits into invariant components with continuous self-similarity.

Full self-similarity requires both parameters in .

Abstract

We exhibit two distinct renormalization scenarios in many-parameter families of piecewise isometries (PWI) of a rhombus. The rotational component, defined over the quadratic field $\mathbb{K}=\mathbb{Q}(\sqrt{5})$, is fixed. The translations are specified by affine functions of the parameters, with coefficients in $\mathbb{K}$. In each case the parameters range over a convex domain. In one scenario the PWI is self-similar if and only if one parameter belongs to $\mathbb{K}$, while the other is free. Such a continuous self-similarity is due to the possibility of merging adjacent atoms of an induced PWI, a common phenomenon in the Rauzy-Veech induction for interval exchange transformations. In the second scenario, the phase space splits into several disjoint (non-convex) invariant components. We show that each component has continuous self-similarity, but due to the transversality of the corresponding foliations, full self-similarity in phase space is achieved if and only if both parameters belong to $\mathbb{K}$. All our computations are exact, using algebraic numbers.