On Aubry's completeness conjecture

TL;DR

This paper proves Aubry's conjecture that the graph of rotation numbers for a twist map is a singular function called a complete devil's staircase under hyperbolic minimal configurations, with implications for physical models.

Contribution

It establishes the conditions under which Aubry's completeness conjecture holds or fails, linking hyperbolic dynamics to the structure of rotation number functions.

Findings

Complete devil's staircase occurs with hyperbolic minimal configurations.

In the presence of KAM tori, the devil's staircase is incomplete.

Results connect dynamical systems to physical phenomena like insulators and quantum effects.

Abstract

In this paper, we prove Aubry's completeness stating conjecture that for a twist map the graph of rotation numbers as a function of the cohomology classes is a purely singularly continuous function (called complete devil's staircase by Aubry) when the set of all minimal configurations is uniformly hyperbolic. Such a phenomenon is crucial for characterizing the chain of atoms being an insulator for the Frenkel-Kontorova model, and can be considered as the analogue of the phase locking phenomenon in critical circle maps as well as the fractional quantum Hall effect. In contrast, in the presence of a positive measure set of KAM tori, we prove that the devil's staircase is incomplete.