On the Novikov problem for dihedral symmetry potentials

TL;DR

This paper investigates the behavior of level lines of quasiperiodic functions with dihedral symmetry potentials, revealing they can only have open level lines at a specific energy, resembling random potentials.

Contribution

It demonstrates that dihedral symmetry quasiperiodic potentials have open level lines only at a single energy level, providing new insights into their geometric properties.

Findings

Open level lines occur only at a specific energy level.

Dihedral symmetry potentials resemble random potentials in behavior.

Level lines are restricted to a single energy, unlike more complex cases.

Abstract

We consider Novikov's problem of describing level lines of quasiperiodic functions on a plane for two-dimensional potentials of dihedral symmetry. It is shown that quasiperiodic potentials of this type can have open level lines only at a single energy level $\, \epsilon = \epsilon_{0} \, $, which brings them close to random potentials on a plane.