与分形相关的新普适类

中文摘要 AI 辅助

正交多项式零点的局部行为由Christoffel-Darboux（CD）核的局部标度行为决定。此前所有研究的行为均由标度极限描述，不同的极限核对应不同的普适类。本文描述了新的普适类，其中不存在单一的极限核，而是存在一个极限环，这类普适类自然适用于康托谱和测度的分形行为。我们证明，这些新现象在两类具有奇异测度的典型模型中出现：三等分康托测度，以及扩张多项式实Julia集上的平衡测度。特别地，这是关于具有奇异谱的几乎周期算子的CD核局部行为的首个结果。作为补充结果，我们描述了二次迭代不动点处的标度函数和Christoffel函数的渐近行为，这是对具有奇异谱的几乎周期模型的首次此类分析。由此我们得出结论：在该不动点处，n次多项式零点的局部标度恰好为n^{-1/α}阶，其中α是测度的局部维数。我们还研究了该情况下的极限链，证明其可通过关于马丁函数（所谓M型）的渐近行为参数化，这是具有奇异测度的链的此类首个结果。

英文摘要

The local behavior of zeros of orthogonal polynomials is determined by the local scaling behavior of Christoffel-Darboux (CD) kernels. All previously studied behaviors are described by scaling limits, and different values of the limit kernel correspond to different universality classes. In this paper, we describe new universality classes in which instead of a single limit kernel, there is a limit cycle. These are naturally suited to Cantor spectra and to fractal behaviors of the measure. We show that these new phenomena occur for two canonical models with singular measures: the middle third Cantor measure and the balanced/equilibrium measure on a real Julia set of an expanding polynomial. In particular, this is the first result on the local behavior of CD kernels for an almost periodic operator with singular spectrum. As a complementary result, we describe the asymptotics of the scaling function, and the Christoffel function, at a fixed point of a quadratic iteration. This is the first such analysis for an almost periodic model with singular spectrum. It allows us to conclude that, at the fixed point, the local scaling of zeros of the polynomial of degree $n$ is precisely of order $n^{-1/α}$, where $α$ is the local dimension of the measure. We also study the limit chain in this case, and prove that it can be parametrized by its asymptotics with respect to a Martin function (so-called $M$-type); this is the first result of this kind for a chain with a singular measure.

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63 pages