Ergodicity in Golden Billiards

TL;DR

This paper analyzes the ergodic properties of irrational right triangular billiards, especially those with angles related to the golden ratio, discussing evidence for nonergodicity and proposing explanations consistent with KAM theory.

Contribution

It provides an interpretation of numerical results on golden billiards and discusses conditions under which these systems may be ergodic, bridging gaps in previous understanding.

Findings

Numerical evidence suggests nonergodicity in certain irrational billiards.

Discussion of how KAM theory may still apply to these systems.

Proposes a reconciliation of ergodic behavior with existing theoretical frameworks.

Abstract

This comment is an analysis of the results presented by Wang et al. in their their 2014 paper on irrational right triangular billiards. They submit numerical evidence that these billiards are a novel kind of nonergodic, incompatible with KAM theory, at least in the "strongly irrational" case, typified by one of the angles being defined by the golden ratio. We offer an explanation of their results as well as a discussion of ergodicity. We suggest that the system is likely to be ergodic and offer a way to reconcile it with KAM theory if it is not.