A first integrability result for Miquel dynamics

TL;DR

This paper proves the integrability of Miquel dynamics for certain biperiodic circle patterns by linking it to elliptic curve translation, using geometric insights into quartic curves.

Contribution

It provides the first integrability result for Miquel dynamics by connecting it to elliptic curve translation through geometric interpretation.

Findings

Miquel dynamics corresponds to translation on an elliptic curve for specific patterns.

The paper introduces a geometric interpretation of addition law on quartic curves.

First proof of integrability for this class of Miquel dynamics.

Abstract

Miquel dynamics is a discrete-time dynamical system on the space of square-grid circle patterns. For biperiodic circle patterns with both periods equal to two, we show that the dynamics corresponds to translation on an elliptic curve, thus providing the first integrability result for this dynamics. The main tool is a geometric interpretation of the addition law on the normalization of binodal quartic curves.