Spectral gaps without the pressure condition

On spectral gaps of Markov maps

It is shown that if a Markov map $T$ on a noncommutative probability space $\mathcal{M}$ has a spectral gap on $L_2(\mathcal{M})$, then it also has one on $L_p(\mathcal{M})$ for $1<p<\infty$. For fixed $p$, the converse also holds if $T$ is factorizable. These results are also new for classical probability spaces.

具有重叠的自共形集的谱间隙与 Fourier 维数

Spectral gaps and Fourier dimension for self-conformal sets with overlaps

本文证明满足 UNI 的重叠 C² 迭代函数系统在临界线附近具有一致谱间隙，并由此推出无原子自共形测度的 Fourier 变换多项式衰减及任意重叠下的分形不确定性原理。

Discrete Uniqueness Sets for Functions with Spectral Gaps

It is well-known that entire functions whose spectrum belongs to a fixed bounded set $S$ admit real uniformly discrete uniqueness sets $Λ$. We show that the same is true for much wider spaces of continuous functions. In particular, Sobolev spaces have this property whenever $S$ is a set of infinite measure having "periodic gaps". The periodicity condition is crucial. For sets $S$ with randomly distributed gaps, we show that the uniformly discrete sets $Λ$ satisfy a strong non-uniqueness property: Every discrete function $c(λ)\in l^2(Λ)$ can be interpolated by an analytic $L^2$-function with sp…

Fourier dimension and spectral gaps for hyperbolic surfaces

We obtain an essential spectral gap for a convex co-compact hyperbolic surface $M=Γ\backslash\mathbb H^2$ which depends only on the dimension $δ$ of the limit set. More precisely, we show that when $δ>0$ there exists $\varepsilon_0=\varepsilon_0(δ)>0$ such that the Selberg zeta function has only finitely many zeroes $s$ with $\Re s>δ-\varepsilon_0$. The proof uses the fractal uncertainty principle approach developed by Dyatlov-Zahl [arXiv:1504.06589]. The key new component is a Fourier decay bound for the Patterson-Sullivan measure, which may be of independent interest. This bound uses the fac…

平面中自共形测度的谱间隙与傅里叶衰减

Spectral gaps and Fourier decay for self-conformal measures in the plane

本文在平面自共形迭代函数系满足温和非线性和不可约性条件下，证明转移算子的均匀谱间隙估计，并由此建立自共形测度的多项式傅里叶衰减，且无需分离性或 Federer 性质。

复杂周期系统中的层级带隙

Hierarchical Band Gaps in Complex Periodic Systems

本文提出复杂周期系统可从其晶胞组成元素继承谱带隙的理论，证明若某频率在所有组成元素中均处于带隙，则必在整体结构中处于带隙，从而无需数值模拟即可预测大晶胞周期系统的谱特征，并通过振动声学与力学示例验证其有效性。