Artur Avila
We prove that the spectrum of the almost Mathieu operator is absolutely continuous if and only if the coupling is subcritical. This settles Problem 6 of Barry Simon's list of Schrödinger operator problems for the twenty-first century.
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We show that the integrated density of states of the almost Mathieu operator is absolutely continuous if and only if the coupling is non-critical. We deduce for subcritical coupling that the spectrum is purely absolutely continuous for almost every phase, settling the measure-theoretical case of Problem 6 of Barry Simon's list of Schrödinger operator problems for the twenty-first century.
It is known that the spectral type of the almost Mathieu operator depends in a fundamental way on both the strength of the coupling constant and the arithmetic properties of the frequency. We study the competition between those factors and locate the point where the phase transition from singular continuous spectrum to pure point spectrum takes place, which solves Jitomirskaya's conjecture in \cite{Ji95,J07}. Together with \cite{Aab}, we give the sharp description of phase transitions for the almost Mathieu operator.
Twenty dry Martinis for the Unitary Almost Mathieu Operator
本文在非临界区域中，针对具有丢番图频率的酉几乎 Mathieu 算子，解决了干十马提尼问题。
Exact local distribution of the absolutely continuous spectral measure
该研究拓展了单频薛定谔算子谱测度连续性的相关结论，精确刻画了稠密小势场下谱测度的局部分布，并探究了其在次临界能量下的分层赫尔德连续性。
The Sharp Spectral Transition for Almost Mathieu Operators via Alternating Resonances
该研究通过交替共振方法，证明了几乎Mathieu算子在特定条件下呈现Anderson局域化，解决了相关文献中的猜想。
In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, \[ (H_{b,ϕ} x)_n= x_{n+1} +x_{n-1} + b \cos(2 πn ω+ ϕ)x_n \] on $l^2(\mathbb{Z})$ and its associated eigenvalue equation to deduce that for $b \ne 0,\pm 2$ and $ω$ Diophantine the spectrum of the operator is a Cantor subset of the real line. This solves the so-called ``Ten Martini Problem'' for these values of $b$ and $ω$. Moreover, we prove that for $|b|\ne 0$ small enough or large enough all spectral gaps predicted by the Gap Labelling theorem are open.
Cantor Spectrum via a Reducibility-Duality Bridge for the Mosaic Almost Mathieu Operator
针对镶嵌准Mathieu算子，通过建立对偶框架并将态密度积分与纤维旋转数的对应关系推广到该模型，结合Aubry对偶、可约性理论和Moser-Pöschel论证，证明了所有非临界参数下谱为Cantor集。
We establish sharp results on the modulus of continuity of the distribution of the spectral measure for one-frequency Schrodinger operators with Diophantine frequencies in the region of absolutely continuous spectrum. More precisely, we establish 1/2-Holder continuity near almost reducible energies (an essential support of absolutely continuous spectrum). For non-perturbatively small potentials (and for the almost Mathieu operator with subcritical coupling), our results apply for all energies.
We consider skew-product maps over circle rotations $x\mapsto x+α$ (mod 1) with factors that take values in SL(2,R). This includes maps of almost Mathieu type. In numerical experiments, with $α$ the inverse golden mean, Fibonacci iterates of maps from the almost Mathieu family exhibit asymptotic scaling behavior that is reminiscent of critical phase transitions. In a restricted setup that is characterized by a symmetry, we prove that critical behavior indeed occurs and is universal in an open neighborhood of the almost Mathieu family. This behavior is governed by a periodic orbit of a renormal…
The absolutely continuous spectrum of an ergodic family of one-dimensional Schrödinger operators is completely determined by the Lyapunov exponent as shown by Ishii, Kotani and Pastur. Moreover, the part of the theory developed by Kotani gives powerful tools for proving the absence of absolutely continuous spectrum, the presence of absolutely continuous spectrum, and even the presence of purely absolutely continuous spectrum. We review these results and their recent applications to a number of problems: the absence of absolutely continuous spectrum for rough potentials, the absence of absolute…
Generalized Aubry-André formula and continuity of the intersection spectrum of the Almost Mathieu operator
本文研究 Almost Mathieu 运算符的谱，证明了 Lebesgue 测度在交集谱上的矩是耦合常数 λ 的多项式，系数为频率 α 的三角多项式，推广了 Aubry-André 公式，并证明了测度在参数下的连续性及分析性。
Computer Validation of Open Gaps for the Almost Mathieu Operator with Critical Coupling
本研究提出动力学与谱两种计算机辅助方法，验证临界耦合下Almost Mathieu算子的开间隙存在性并估计大小，给出不同ω下的开间隙数量结果，还提出相关周期问题的实验与猜想。
We show that for almost every frequency alpha \in \R \setminus \Q, for every C^omega potential v:\R/\Z \to R, and for almost every energy E the corresponding quasiperiodic Schrodinger cocycle is either reducible or nonuniformly hyperbolic. This result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic Schrodinger operator, and allows us to complete the proof of the Aubry-Andre conjecture on the measure of the spectrum of the Almost Mathieu Operator.
We prove for the square Fibonacci Hamiltonian that the density of states measure is absolutely continuous for almost all pairs of small coupling constants. This is obtained from a new result we establish about the absolute continuity of convolutions of measures arising in hyperbolic dynamics with exact-dimensional measures.
We prove the conjecture (known as the ``Ten Martini Problem'' after Kac and Simon) that the spectrum of the almost Mathieu operator is a Cantor set for all non-zero values of the coupling and all irrational frequencies.
We study the phase transion line of the almost Mathieu operator, that separates arithmetic regions corresponding to singular continuous and a.e. pure point regimes, and prove that both purely singular continuous and a.e. pure point spectrum occur for dense sets of frequencies.
Subordinacy theory for long-range operators: hyperbolic geodesic flow insights and monotonicity theory
本文提出了适用于长程算子的从属理论框架，通过结合双曲测地流几何洞察与无坐标单调性理论，解决了有限程算子绝对连续谱的开放问题，并首次证明了准周期长程算子在全相位下的纯绝对连续谱持久性。
We develop some non-perturbative methods for studying the IDS in almost Mathieu and related models. Assuming positive Lyapunov exponents, and assuming that the potential function is a trigonometric polynomial of degree k, we show that the Holder exponent of the IDS is almost 1/2k. We also show that this is stable under small perturbations of the potential (e.g., potentials which are close to that of almost Mathieu again give rise to almost 1/2 Holder continuous IDS). Moreover, off a set of Hausdorff dimension zero the IDS is Lipschitz. We further deduce from these properties that the IDS is ab…
Positive-Measure Acceleration Strata for Prevalent Perturbations of N-Fold Pullbacks of Supercritical Almost Mathieu Potentials
本文证明在超临界几乎Mathieu势的N重拉回中，对常见的实解析扰动，存在正测度的谱集，其上加速等于整数s（1≤s≤N），通过首次返回余循环的零点重数、Poisson积分与Kac归一化及Schur变分实现。
Quasiperiodic tilings are often considered as structure models of quasicrystals. In this context, it is important to study the nature of the diffraction measures for tilings. In this article, we investigate the diffraction measures for S-adic tilings in R^d, which are constructed from a family of geometric substitution rules. In particular, we firstly give a sufficient condition for the absolutely continuous component of the diffraction measure for an S-adic tiling to be zero. Next, we prove this sufficient condition for "almost all" binary block-substitution cases and thus prove the absence o…
A class of piecewise affine hyperbolic maps on a bounded subset of the plane is considered. It is shown that if a map from this class is sufficiently area-expanding then almost surely this map has an absolutely continuous invariant measure.
We show that for entire maps of the form $z \mapsto λ\exp(z)$ such that the orbit of zero is bounded and such that Lebesgue almost every point is transitive, no absolutely continuous invariant probability measure can exist. This answers a long-standing open problem.
For non-critical almost Mathieu operators with Diophantine frequency, we establish exponential asymptotics on the size of spectral gaps, and show that the spectrum is homogeneous. We also prove the homogeneity of the spectrum for Schödinger operators with (measure-theoretically) typical quasi-periodic analytic potentials and fixed strong Diophantine frequency. As applications, we show the discrete version of Deift's conjecture \cite{Deift, Deift17} for subcritical analytic quasi-periodic initial data and solve a series of open problems of Damanik-Goldstein et al \cite{BDGL, DGL1, dgsv, Go} and…
We prove that for certain partially hyperbolic skew-products, non-uniform hyperbolicity along the leaves implies existence of a finite number of ergodic absolutely continuous invariant probability measures which describe the asymptotics of almost every point. The main technical tool is an extension for sequences of maps of a result of de Melo and van Strien relating hyperbolicity to recurrence properties of orbits. As a consequence of our main result, we also obtain a partial extension of Keller's theorem guaranteeing the existence of absolutely continuous invariant measures for non-uniformly …
We prove that the quasi-periodic Schrödinger operator with a finitely differentiable potential has purely absolutely continuous spectrum for all phases if the frequency is Diophantine and the potential is sufficiently small in the corresponding $C^k$ topology. This is based on a refined quantitative $C^{k,k_0}$ almost reducibility theorem which only requires a quite low initial regularity ``$k>14τ+2$'' and much of the regularity ``$k_0\leq k-2τ-2$'' is conserved in the end, where $τ$ is the Diophantine constant of the frequency.
Sharp Spectral Gaps, Arithmetic Localization, and Reducibility via Resonance Analys
本文通过共振分析，研究了准周期Schrödinger算子的谱间隙、算术定位及可约性问题，解决了多个开放性猜想并揭示了亚临界算子的分层结构。
We prove existence of (at most denumerable many) absolutely continuous invariant probability measures for random one-dimensional dynamical systems with asymptotic expansion. If the rate of expansion (Lyapunov exponents) is bounded away from zero, we obtain finitely many ergodic absolutely continuous invariant probability measures, describing the asymptotics of almost every point. We also prove a similar result for higher-dimensional random non-uniformly expanding dynamical systems. The results are consequences of the construction of such measures for skew-products with essentially arbitrary ba…
Absolutely continuous invariant measures and the Collet-Eckmann condition for long-branched Sturmian unimodal maps
该研究聚焦长分支斯特姆单峰映射，通过揉类与特定洛伦兹构造的关联，计算切割共切割时间，建立绝对连续不变测度判据，还证明几乎所有旋转角下无有限阶S-单峰实现满足Collet-Eckmann条件。
Cantor spectrum for multidimensional quasi-periodic Schrödinger operators
研究多维准周期Schrödinger算子的Cantor谱问题，证明临界近正交算子的谱Hausdorff维数可任意小但非零。
Smoothness of random self-similar measures on the line and the existence of interior points
该研究探讨直线上随机自相似测度密度的光滑性，证明符号局部维数处处大于1的测度的自然投影几乎必然绝对连续且密度Hölder连续，还得出相似维数大于1时随机自相似集几乎必然含内点的结论。
Hyperbolic absolutely continuous invariant measures for C^r one-dimensional maps
本文证明，对 r>1 的 C^r 区间或圆周映射，若在正测度集上 Lyapunov 指数超过阈值，则存在关于 Lebesgue 测度绝对连续的双曲遍历不变测度，且其吸引域几乎处处覆盖该集合。
We prove uniform hyperbolicity of the renormalization operator for all possible real combinatorial types. We derive from it that the set of infinitely renormalizable parameter values in the real quadratic family $P_c: x\mapsto x^2+c$ has zero measure. This yields the statement in the title (where ``regular'' means to have an attracting cycle and ``stochastic'' means to have an absolutely continuous invariant measure). An application to the MLC problem is given.
Universal Weighted Averaging for Ergodic Flows
本文提出了一种用于遍历流的通用加权平均方法，并通过实例证明了刚性流不具备几乎混合性质。
Finitude of physical measures for random maps
本文利用 Markov 算子刻画随机映射的绝对连续遍历平稳测度，建立物理测度有限且吸引盆几乎覆盖全空间的层级与充分条件。
Since the pioneering works of Jakobson and Benedicks & Carleson and others, it has been known that a positive measure set of quadratic maps admit invariant probability measures absolutely continuous with respect to Lebesgue. These measures allow one to statistically predict the asymptotic fate of Lebesgue almost every initial condition. Estimating fluctuations of empirical distributions before they settle to equilibrium requires a fairly good control over large parts of the phase space. We use the sub-exponential slow recurrence condition of Benedicks & Carleson to build induced Markov maps of…
Natural measures and statistical properties of non-statistical maps with multiple neutral fixed points
本文研究了不具有物理测度的无限测度保持动力系统，证明了其具有强统计性质，并给出了自然测度存在的充分条件及经验测度的分布极限定律。
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