Xin Zhao
We establish sharp results on the modulus of continuity of the distribution of the spectral measure for multi-frequency Schrodinger operators with Diphantine frequencies and small analytic potentials.
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We prove the Hölder continuity of the Lyapunov exponent for quasi-periodic Schrödinger cocycles with a $C^2$ cos-type potential and any fixed Liouvillean frequency, provided the coupling constant is sufficiently large. Moreover, the Hölder exponent is independent of the frequency and the coupling constant.
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.
For Hölder continuous cocycles over an invertible, Lipschitz base, we establish the Hölder continuity of Oseledets subspaces on compact sets of arbitrarily large measure. This extends a result of Araújo, Bufetov, and Filip by considering possibly noninvertible cocycles, which in addition may take values in the space of compact operators on a Hilbert space. As a by-product of our work, we also show that a noninvertible cocycle with nonvanishing Lyapunov exponents exhibits nonuniformly hyperbolic behaviour (in the sense of Pesin) on a set of full measure.
We show that if the base frequency is Diophantine, then the Lyapunov exponent of a $C^{k}$ quasi-periodic $SL(2,\mathbb{R})$ cocycle is $1/2$-Hölder continuous in the almost reducible regime, if $k$ is large enough. As a consequence, we show that if the frequency is Diophantine, $k$ is large enough, and the potential is $C^k$ small, then the integrated density of states of the corresponding quasi-periodic Schrödinger operator is $1/2$-Hölder continuous.
Hölder continuity of core entropy for non-recurrent quadratic polynomials
本文通过对Mandelbrot集和Hubbard树动力学的符号分析，证明了一大类关于角倍增非回归的二次多项式的核熵作为外角的函数是Hölder连续的。
Hölder continuity of measures for heavy tail potentials
本文针对满足特定条件的重尾势，证明了测度的 EKP 不等式，即弱*范数下关于压力的 Hölder 连续性，并揭示了间歇流的新相变。
We consider a similar type of scenario for the disappearance of uniform of hyperbolicity as in Bjerklöv and Saprykina (2008, Nonlinearity 21), where it was proved that the minimum distance between invariant stable and unstable bundles has a linear power law dependence on parameters. In this scenario we prove that the Lyapunov exponent is sharp $\frac12$-Hölder continuous. In particular, we show that the Lyapunov exponent of Schrödinger cocycles with a potential having a unique non-degenerate minimum, is sharp $\frac12$-Hölder continuous below the lowest energy of the spectrum, in the large cou…
For a large class of nonuniformly expanding maps of $\Bbb R^m$, with indifferent fixed points and unbounded distorsion and non necessarily Markovian, we construct an absolutely continuous invariant measure. We extend to our case techniques previously used for expanding maps on quasi-Hölder spaces. We give general conditions and provide examples to which apply our result.
Hölder continuity of Oseledets subspaces for linear cocycles on Banach spaces
本文研究了Banach空间上线性余切的Oseledets子空间，在可逆与不可逆情形下证明了其Hölder连续性，推广了已有结果。
We consider the quasi-periodic Jacobi operator $H_{x,ω}$ in $l^2(\mathbb{Z})$ $(H_{x,ω}ϕ)(n) = -b(x+(n+1)ω)ϕ(n+1) - b(x+nω)ϕ(n-1) + a(x+nω)ϕ(n) = Eϕ(n),\ n\in\mathbb{Z},$ where $a(x),\ b(x)$ are analytic function on $\mathbb{T}$, $b$ is not identically zero, and $ω$ obeys some strong Diophantine condition. We consider the corresponding unimodular cocycle. We prove that if the Lyapunov exponent $L(E)$ of the cocycle is positive for some $E=E_0$, then there exists $ρ_0=ρ_0(a,b,ω,E_0)$, $β=β(a,b,ω)$ such that $|L(E)-L(E')|<|E-E'|^β$ for any $E,E'\in (E_0-ρ_0,E_0+ρ_0)$. If $L(E)>0$ for all $E$ in …
On the existence of invariant absolutely continuous probability measures for $C^1$ expanding maps of the circle
本文针对圆周 $C^1$ 一致扩张映射，构造了具有指定最优连续模及绝对连续不变概率测度或保持 Lebesgue 测度的不可数多例，并证明其中许多映射具有无界畸变。
We show how to turn the question of the absolute continuity of Bernoulli convolutions into one of counting the growth of the number of overlaps in the system. When the contraction parameter is a hyperbolic algebraic integer, we turn this question of absolute continuity into a question involving the ergodic theory of cocycles over domain exchange transformations.
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。
We study the spectral location of strongly pattern equivariant Hamiltonians arising through configurations on a colored lattice. Roughly speaking, two configurations are "close to each other" if, up to a translation, they "almost coincide" on a large fixed ball. The larger this ball is, the more similar they are, and this induces a metric on the space of the corresponding dynamical systems. Our main result states that the map which sends a given configuration into the spectrum of its associated Hamiltonian, is Hölder (even Lipschitz) continuous in the usual Hausdorff metric. Specifically, the …
Hölder continuity and laminarity of the Green currents for Hénon-like maps
研究Hénon-like映射的Green流的Hölder连续超势及层性，证明其最大熵测度为Monge-Ampère流，并具有正Hausdorff维数。
Given a hyperbolic homeomorphism on a compact metric space, consider the space of linear cocycles over this base dynamics which are Hölder continuous and whose projective actions are partially hyperbolic dynamical systems. We prove that locally near any typical cocycle, the Lyapunov exponents are Hölder continuous functions relative to the uniform topology. This result is obtained as a consequence of a uniform large deviations type estimate in the space of cocycles. As a byproduct of our approach, we also establish other statistical properties for the iterates of such cocycles, namely a centra…
Anderson localization and Hölder continuity of the integrated density of states for analytic quasiperiodic Schrödinger operators
本文研究了准周期薛定谔算子在扰动范围内，对任意非常数解析势和任意迪ophantine频率的安德森局域化及积分态密度的霍尔德连续性，提出了一种新的格林函数控制方法以消除双重共振。
We prove that a self similar measure is absolutely continuous providing that it satisfies a condition depending on its Garsia entropy, contraction ratio, and the separation between different points in approximations of the self similar measure. In the special case of Bernoulli convolutions this paper gives a sufficient condition for the Bernoulli convolution with parameter lambda to be absolutely continuous in terms of lambda and its Mahler measure.
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.
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