On the Novikov problem for dihedral symmetry potentials

TL;DR

This work studies the Novikov problem for quasiperiodic plane potentials with dihedral symmetry $D_n$ ($n≥3$). Using a completely irrational embedding of the plane into $\mathbb{R}^N$ and phase transformations, the authors show that open level lines are chaotic and can appear only at a single energy $ε_0$, with the energy interval collapsing to the point $\{ε_0\}$. For $ε≠ε_0$, all level lines are closed and their maximal size $D(ε)$ grows as $ε\toε_0$, highlighting the proximity of these dihedral-symmetric potentials to random two-dimensional potentials and their relevance to two-dimensional quasicrystals. The results illuminate how the geometry of open trajectories connects to energy emergence in the generalized Novikov problem and may inform broader applications in quasicrystal theory.

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 $\, ε= ε_{0} \, $, which brings them close to random potentials on a plane.

Figures (10)

• Figure 1: General form of a "topologically regular" open level line of a quasiperiodic potential on a plane.
• Figure 2: General form of a "chaotic" level line of a quasiperiodic potential on a plane.
• Figure 3: Symmetry axes of the potential $\, V (x, y) \,$, passing through one point in the plane $\, \Pi \,$ and dividing the plane into $\, 2 n \,$ sectors $\, S_{i}$.
• Figure 4: Possible form of open level lines (\ref{['E12Lines']}) (schematically).
• Figure 5: Curves $\, \gamma_{1} \,$ and $\, \gamma_{2} \,$ lying at levels $\, \epsilon = E_{1} \,$ and $\, \epsilon = E_{2} \,$ (in each of the sectors $\, S_{i} \,$).