“分形几何及应用” 学术报告会


学 术 报 告1

题目: Inner functions, Fatou’s exceptional sets, and composition operators on the weighted Bergan space

报告人:董新汉(湖南师范大学)

时间:2020年11月28日8:30-10:00

地点: 腾讯会议 会议 ID:279 325 766

报告摘要: Let Cφ : f → f ◦ φ be a linear composition operator on the weighted Bergman spaces Ap α . From Fatou theorem, limr→1− φ(reix) exists almost everywhere, and we call this set, which is removed by a.e., an exceptional set, and denote it by Exp(φ). It follows from Shapiro Theorem that Exp(φ) is a zero and dense set on ∂D if Cφ is compact and φ is an inner function. In the first part of this talk, we consider the singular inner function φ := φµ which depends on a singular positive measure µ on [0, 1]. By using the language of measure theory, we establish four equivalent conditions for the compactness of Cφ; further, for some singular self similar measures and some discrete measures, the necessary and sufficient conditions for the compactness of Cφ are obtained. In the second part, we give a relationship between the Hausdorff dimensions dimHExp(φ) of Fatou’s exceptional set Exp(φ) and the compactness of Cφ for some inner functions φ, this study involves Diophantine theory in continued fractions and O. Frostman’s Theorem and M. Riesz’s Theorem for the inner function.

报告人简介:董新汉,男,博士学位(CUHK,导师Ka-Sing Lau/Shing-Tung Yau),湖南师范大学数学二级教授,博士生导师,享受国务院政府特殊津贴专家。曾任湖南师范大学数学与计算机科学学院院长(2002.10-2012.10),国家自然科学基金委重点项目、面上项目、青年项目会评专家,国家杰出青年基金会评专家,国家自然科学奖数学组会评专家,教育部高等学校数学与统计学教学指导委员会委员,湖南省数学学会副理事长。获湖南省省级教学成果一等奖,所指导的研究生获湖南省优秀博士论文、湖南省杰出青年基金、湖南省优秀青年基金。 数学研究兴趣: 复分析方向与分形几何方向。研究成果在国际著名数学期刊Adv. Math., Trans. Amer. Math. Soc., J. Funct. Anal.等杂志上发表。主持国家自然科学基金项目7项,其中重点项目1项(在研期:2019.1-2023.12)




学 术 报 告2

题目: Spectrality of Moran-Sierpinski Measures

报告人:邓启荣(福建师范大学)

时间:2020年11月28日10:20-11:50

地点: 腾讯会议 会议 ID:279 325 766

报告摘要: Let $A_n\in M_2(\Z)$ be integral matrices such that the infinite convolution of Dirac measures with equal weights

$$\mu_{\{A_n, n\ge 1\}}:=\delta_{A_1^{-1}\D}\ast\delta_{A_1^{-1}A_2^{-2}\D}\ast\cdots$$

is a probability measure with compact support, where $\D=\{0, (1, 0)^t, (0, 1)^t\}$ is the Sierpinski digit. We prove that there exists a set $\Lambda\subset\R^2$ such that the family $\{e^{2\pi i <\lambda,x>}: \lambda\in\Lambda\}$ is an orthonormal basis of $L^2( \mu_{\{A_n, n\ge 1\}})$ if and only if $\frac 13(1, -1)A_n\in\Z^2$ for $n\ge 2$ under some metric conditions on $A_n$.

报告人简介: 福建师范大学数学系教授、博士生导师。1987年获得云南大学数学系硕士学位(统计方向),2005年获香港中文大学数学系博士学位(分形几何方向),最近十年从事分形几何及相关问题研究,在Journal of Functional Analysis, Nonlinearity等国际著名期刊发表论文20多篇,正主持国家自然科学基金面上项目一项。




学 术 报 告3

题目: Arbitrarily sparse spectra for self-affine spectral measures

报告人:安丽想(华中师范大学)

时间:2020年11月28日14:30-16:00

地点: 腾讯会议 会议 ID:279 325 766

报告摘要: Given an expansive matrix R Md(Z) and a finite set of digit B taken from Zd=R(Zd). It was shown previously that if we can find an L such that (R; B; L) forms a Hadamard triple, then the associated fractal self-a_ne measure generated by (R; B) admits an exponential orthonormal basis of certain frequency set _, and hence it is termed as a spectral measure. We show that if #B < j det(R)j, not only it is spectral, we can also construct arbitrarily sparse spectrum _ in the sense that its Beurling dimension is zero.This is based on joint work with Chunkit Lai.

报告人简介: 博士,华中师范大学教授,主要研究方向为测度论及其应用、分形几何、小波分形、调和与非调和Fourier分析。研究成果在国际著名数学期刊Adv. Math., Trans. Amer. Math. Soc., J. Funct. Anal.等杂志上发表




学 术 报 告4

题目: Several problems on the convergence of Fourier series w.r.t. spectral measures.

报告人:何兴纲(华中师范大学)

时间:2020年11月29日8:30-10:00

地点: 腾讯会议 会议 ID:920 305 296

报告摘要: In this talk we will review some known results on the convergence or divergence of Fourier series with respect to spectral measures and arise several problems on this topic.

报告人简介: 何兴纲,华中师范大学教授,博士生导师。主要研究方向为测度论及其应用、 调和与非调和Fourier分析、分形几何、小波分析等。相关成果已发表于Adv. Math, J. Funct.Anal., J. Math. Pures Appl.等国际期刊。




学 术 报 告5

题目: Open set condition and pseudo Hausdorff measure of self-affine IFSs

报告人:付小叶(华中师范大学)

时间: 2020年11月28日10:20-11:50

地点: 腾讯会议 会议 ID:920 305 296

报告摘要: Let $A$ be an $n\times n$ real expanding matrix and $\mathcal{D}$ be a finite subset of $\mathbb{R}^n$

with $0\in\mathcal{D}$. The family of maps $\{f_d(x)=A^{-1}(x+d)\}_{d\in\mathcal{D}}$ is called a self-affine iterated function system (self-affine IFS). The self-affine set $K=K(A,\mathcal{D})$ is the unique compact set determined by $(A, {\mathcal D})$ satisfying the set-valued equation $K=\displaystyle\bigcup_{d\in\mathcal{D}}f_d(K)$.

The number $s=n\,\ln(\# \mathcal{D})/\ln(q)$ with $q=|\det(A)|$, is the so-called pseudo similarity dimension of $K$. As shown by He and Lau, one can associate with $A$ and any number $s\ge 0$ a natural pseudo Hausdorff measure denoted by $\mathcal{H}_w^s.$ In this paper, we show that, if $s$ is chosen to be the pseudo similarity dimension of $K$, then the condition $\mathcal{H}_w^s(K)> 0$ holds if and only if the IFS $\{f_d\}_{d\in\mathcal{D}}$ satisfies the open set condition (OSC). This extends the well-known result for the self-similar case that the OSC is equivalent to $K$ having positive Hausdorff measure $\mathcal{H}^s$ for a suitable $s$.

Furthermore, we relate the exact value of pseudo Hausdorff measure $\mathcal{H}_w^s(K)$ to a notion of upper $s$-density with respect to the pseudo norm $w(x)$ associated with $A$ for the measure

$\mu=\lim\limits_{M\to\infty}\sum\limits_{d_0,\dotsc,d_{M-1}\in\mathcal{D}}\delta_{d_0 + Ad_1 + \dotsb + A^{M-1}d_{M-1}}$ in the case that $\#\mathcal{D}\le\lvert\det A\rvert$.

报告人简介: 付小叶,华中师范大学教授,硕士研究生导师,国家自然科学基金由青获得者。2011年博士毕业于加拿大McMaster大学,主要从事小波分析和分形tile、tiling相关研究;2012-2014年在香港中文大学从事博士后研究,主要从事与Fuglede谱集猜想、测度谱性的相关研究;主要研究领域涉及傅里叶分析、小波分析、分形几何、压缩感知及应用。研究成果发表于Memoirs of AMS, Adv.Math., J. Funct. Anal.,Constr. Approx.等期刊上。




学 术 报 告6

题目: Spectrality of generalized Sierpinski-type self-affine measures

报告人:刘竟成(湖南师范大学)

时间:2020年11月28日14:30-16:00

地点: 腾讯会议 会议 ID:920 305 296

报告摘要: Let µ be a Borel probability measure with compact support on Rn. We call it a spectral measure if there exists a countable subset ʌ such that the family of exponential functions E(ʌ):={e2pi :lÎ ʌ } forms an orthonormal basis for L2(µ).

In this talk, we will consider the spectrality of generalized Sierpinski-type self-affine measures, and give the necessary and sufficient conditions for it to be a spectral measure.

报告人简介: 博士,湖南师范大学教授,硕士生导师。 湖南省自然科学基金杰出青年基金获得者,主要从事分形几何与复分析、函数空间与算子理论的研究。研究成果发表在Adv. Math., JFA等国际著名期刊。




学 术 报 告7

题目: Spectral eigenvalue problems of Fractal spectral measures

报告人:伍智义(中山大学)

时间:2020年11月29日16:20-17:50

地点: 腾讯会议 会议 ID:920 305 296

报告摘要: In this talk we will give a survey on the spectral eigenvalue problems of fractal spectral measures.

报告人简介: 中山大学博士后,主要研究方向为分形几何和调和分析,在国际著名期刊J. Funct. Anal. JFAA等期刊发表论文多篇。