On the almost reducibility conjecture

Avila's Almost Reducibility Conjecture (ARC) is a powerful statement linking purely analytic and dynamical properties of analytic one frequency $SL(2,\mathbb{C})$ cocycles. It is also a fundamental tool in the study of spectral theory of analytic one-frequency Schr\"odinger operators, with many striking consequencies, allowing to give a detailed characterization of the subcritical region. Here we give a proof, completely different from Avilas, for the important case of Schr\"odinger cocycles and non-exponentially approximated frequencies, allowing, in particular, to obtain all the desired spectral consequences.Comment: 20 page

The H\"older continuity of Lyapunov exponents for a class of Cos-type quasiperiodic Schr\"odinger cocycles

In this paper we obtain exact $\frac{1}{2}$-H\"older continuity of the Lyapunov exponents for quasi-periodic Sch\"odinger cocycles with $C^2$ cos-type potentials, large coupling constants, and fixed Diophantine frequency. Moreover, we prove the locally Lipschitz continuity of the Lyapunov exponent for a full measure spectral set. Furthermore, for any given $r$ between $\frac{1}{2}$ to $1$, we can find some energy on the spectrum and on which Lyapunov exponent is exactly $r$-H\"older continuous.Comment: 88 Page

Robust Source-Free Domain Adaptation for Fundus Image Segmentation

Unsupervised Domain Adaptation (UDA) is a learning technique that transfers knowledge learned in the source domain from labelled training data to the target domain with only unlabelled data. It is of significant importance to medical image segmentation because of the usual lack of labelled training data. Although extensive efforts have been made to optimize UDA techniques to improve the accuracy of segmentation models in the target domain, few studies have addressed the robustness of these models under UDA. In this study, we propose a two-stage training strategy for robust domain adaptation. In the source training stage, we utilize adversarial sample augmentation to enhance the robustness and generalization capability of the source model. And in the target training stage, we propose a novel robust pseudo-label and pseudo-boundary (PLPB) method, which effectively utilizes unlabeled target data to generate pseudo labels and pseudo boundaries that enable model self-adaptation without requiring source data. Extensive experimental results on cross-domain fundus image segmentation confirm the effectiveness and versatility of our method. Source code of this study is openly accessible at https://github.com/LinGrayy/PLPB.Comment: 10 pages, WACV202

Hofstadter butterflies and metal/insulator transitions for moir\'e heterostructures

We consider a tight-binding model recently introduced by Timmel and Mele for strained moir\'e heterostructures. We consider two honeycomb lattices to which layer antisymmetric shear strain is applied to periodically modulate the tunneling between the lattices in one distinguished direction. This effectively reduces the model to one spatial dimension and makes it amenable to the theory of matrix-valued quasi-periodic operators. We then study the transport and spectral properties of this system, explaining the appearance of a Hofstadter-type butterfly. For sufficiently incommensurable moir\'e length and strong coupling between the lattices this leads to the occurrence of localization phenomena

Kotani theory, Puig's argument, and stability of The Ten Martini Problem

We solve the ten martini problem (Cantor spectrum with no condition on irrational frequencies, previously only established for the almost Mathieu) for a large class of one-frequency quasiperiodic operators, including nonperturbative analytic neighborhoods of several popular explicit families. The proof is based on the structural analysis of dual cocycles as introduced in [35]. As a part of the proof, we develop several general ingredients of independent interest: Kotani theory, for a class of finite-range operators over general minimal underlying dynamics, making the first step towards and providing a partial solution of the Kotani-Simon problem, simplicity of point spectrum for the same class, and the all-frequency version of Puig's argument.Comment: 59 page

Multiplicative Jensen's formula and quantitative global theory of one-frequency Schr\"odinger operators

We introduce the concept of dual Lyapunov exponents, leading to a multiplicative version of the classical Jensen's formula for one-frequency analytic Schr\"odinger cocycles. This formula, in particular, gives a new proof and a quantitative version of the fundamentals of Avila's global theory \cite{avila}, fully explaining the behavior of complexified Lyapunov exponent through the dynamics of the dual cocycle. In particular, concepts of (sub/super) critical regimes and acceleration are all explained (in a quantitative way) through the duality approach. This leads to a number of powerful spectral and physics applicationsComment: 40 page