Minimality and nonergodicity on a family of flat surfaces in genus 3

TL;DR

This paper demonstrates that a specific family of genus 3 flat surfaces exhibits uncountably many minimal but nonergodic directions, challenging Veech's Dichotomy through combinatorial and irrationality conditions.

Contribution

It introduces a new family of flat surfaces in genus 3 that are minimal but nonergodic, providing explicit examples and conditions that violate Veech's Dichotomy.

Findings

Uncountably many minimal but nonergodic directions on these surfaces

The Arnoux-Yoccoz surface satisfies the conditions

Contradicts Veech's Dichotomy in genus 3 surfaces

Abstract

We prove that a certain family of flat surfaces in genus 3 does not fulfill Veech's Dichotomy. These flat surfaces provide uncountably many minimal but nonergodic directions. The conditions on this family are a combinatorical one and an irrationality condition. The Arnoux-Yoccoz surface fulfills this conditions.