On connectedness of chaotic sections of some 3-periodic surfaces

TL;DR

This paper constructs a specific periodic surface in three-dimensional space where almost all plane sections in a certain direction are connected, addressing a problem related to electron motion in magnetic fields.

Contribution

It introduces a novel Z^{3}-periodic surface with connected plane sections, utilizing the Rips machine algorithm for band complexes.

Findings

Almost all sections in a specific direction are connected

Addresses a problem from Novikov's electron motion theory

Uses the Rips machine algorithm for analysis

Abstract

In the present paper we construct a Z^{3}-periodic surface in R^{3} whose almost all plane sections of a certain direction consist of exactly one connected component. This question originates from a problem of Novikov on the semi- classical motion of an electron in strong magnetic field. Our main tool is the Rips machine algorighm for band complexes.