Symmetric band complexes of thin type and chaotic sections which are not quite chaotic

TL;DR

This paper constructs symmetric, minimal-rank foliated 2-complexes of thin type, leading to examples of 3-periodic surfaces with chaotic plane sections that are not entirely chaotic, exhibiting asymptotic directions due to non-unique ergodicity.

Contribution

It introduces simpler, symmetric examples of thin type complexes with minimal rank, enabling the construction of 3-periodic surfaces with complex, non-uniquely ergodic chaotic sections.

Findings

Constructed symmetric, minimal-rank complexes of thin type.

Demonstrated existence of 3-periodic surfaces with chaotic sections.

Showed sections have asymptotic directions due to non-unique ergodicity.

Abstract

In a recent paper we constructed a family of foliated 2-complexes of thin type whose typical leaves have two topological ends. Here we present simpler examples of such complexes that are, in addition, symmetric with respect to an involution and have the smallest possible rank. This allows for constructing a 3-periodic surface in the three-space with a plane direction such that the surface has a central symmetry, and the plane sections of the chosen direction are chaotic and consist of infinitely many connected components. Moreover, typical connected components of the sections have an asymptotic direction, which is due to the fact that the corresponding foliation on the surface in the 3-torus is not uniquely ergodic.