The spectrum of period-doubling Hamiltonian

TL;DR

This paper investigates the spectral properties of the period-doubling Hamiltonian, revealing a lower bound on its Hausdorff dimension, the existence of unbounded trace orbits, and a detailed gap structure characterization.

Contribution

It introduces an intrinsic coding of the spectrum, providing new insights into its fractal structure and gap labeling, contrasting with previous results on trace orbits.

Findings

Hausdorff dimension exceeds 

Existence of a dense uncountable subset with unbounded trace orbits

Complete characterization of gap structure and labeling

Abstract

In this paper, we show the following: the Hausdorff dimension of the spectrum of period-doubling Hamiltonian is bigger than $\log \alpha/\log 4$, where $\alpha$ is the Golden number; there exists a dense uncountable subset of the spectrum such that for each energy in this set, the related trace orbit is unbounded, which is in contrast with a recent result of Carvalho (Nonlinearity 33, 2020); we give a complete characterization for the structure of gaps and the gap labelling of the spectrum. All of these results are consequences of an intrinsic coding of the spectrum we construct in this paper.