学术报告报告题目一：H-容度及其性质
报 告 人：文志英 教授(清华大学)
报告时间：2019年4月13日（星期六）上午09:00-09:40
报告题目二：Estimates on the Dimension of Self-Similar Measures with Overlaps
报 告 人：丰德军 教授(香港中文大学)
报告时间：2019年4月13日（星期六）上午09:40-10:20
报告题目三：Estimates on the Dimension of Self-Similar Measures with Overlaps
报 告 人：奚李峰 教授(宁波大学)
报告时间：2019年4月13日（星期六）上午10:40-11:20
报告题目四：Some Topology on Self-Affine Necklaces
报 告 人：文胜友 教授(湖北大学)
报告时间：2019年4月13日（星期六）上午11:20-12:00
报告题目五：Full Cylinders and Diophantine Analysis of Beta-Dynamical System in Parameter Space
报 告 人：吴军 教授(华中科技大学)
报告时间：2019年4月13日（星期六）下午14:30-15:10
报告题目六：Rigidity of Cantor Sets from the Point View of Self-Similar Subsets
报 告 人：饶辉 教授(华中师范大学)
报告时间：2019年4月13日（星期六）下午15:10-15:50
报告题目七：The Spectral Eigenmatrix Problems of the Sierpinski Measures
报 告 人：何兴纲 教授(华中师范大学)
报告时间：2019年4月13日（星期六）下午16:10-16:50
报告题目八：Mass Transference Principle from Rectangles to Rectangles
报 告 人：王保伟 教授(华中科技大学)
报告时间：2019年4月13日（星期六）下午16:50-17:30
报告地点：37号楼3A01室
邀 请 人：熊瑛 教授
欢迎广大师生前往！
数学学院
2019年4月11日
报告一摘要：
H-容度作为一种特殊的外测度，具有一些特殊的性质。报告介绍H-容度的各种基本性质，和其它测度的关系以及一些简单应用。
报告二摘要：
In this talk we will present some algorithms for the computation of the lower and upper bounds for the dimension of self-similar measures with overlaps. As examples, we provide some numerical estimates on the dimension of Bernoulli convolutions. This is joint work with Zhou Feng.
报告三摘要：
We identify two fractals if and only if they are bilipschitz equivalent. Fix a ratio r, for dust-like graph-directed sets with ratio r and integer characteristic, we obtain a rigid theorem that these graph-directed sets are uniquely determined by their Hausdorff dimension (or integer characteristic) in the sense of bilipschitz equivalence. Using this rigidity theorem, we show that in a suitable class of self-similar sets, two totally disconnected self-similar sets without complete overlaps are bilipschitz equivalent. We also provide an algorithm to test complete overlaps in polynomial time.
报告四摘要：
The self-affine necklaces introduced in the present paper are a class of connected self-affine sets containing Sierpinski's gaskets. For every self-affine necklace with the property P we prove that all of its NIFSs are of the same cardinality. Moreover, if the necklace is simple, we prove that it has a unique NIFS up to certain permutations and affine maps. In addition, we prove that two simple self-affine necklaces with the property P admit only rigid homeomorphisms and thus admit at most countably many homeomorphisms.
报告五摘要：
Let $\{x_n\}_{n\ge1}\subseteq[0,1]$ be a sequence of real numbers and $\varphi\colon\mathbb N\to(0,1]$ be a positive function. We show that for any $x\in(0,1]$, the set\[ \Bigl\{\beta>1\colon|T_\beta^n x-x_n|<\varphi(n)\ \text{for infinitely many $n\in\mathbb N$}\Bigr\} \]is of zero or full Lebesgue measure in $(1,+\infty)$ according to $\sum\varphi(n)<+\infty$ or not, where $T_\beta$ is the beta-transformation. We also determine, for any $x\in(0,1]$, the exact Lebesgue measure of the set\[ \Bigl\{\beta>1\colon|T_\beta^n x-x_n|<\beta^{-l_n}\ \text{for infinitely many $n\in\mathbb N$}\Bigr\}, \]where $\{l_n\}_{n\ge1}$ is a sequence of nonnegative real numbers.
报告六摘要：
Let $L\geq 2$ be an integer and let $0<\alpha<1/L^2$. Let $\mathcal C_{\alpha,L}$ be the uniform Cantor set defined by the following set equation$$\mathcal C_{\alpha,L}= \bigcup_{j=0}^{L-1} \alpha(C_{\alpha,L}+j).$$We show that for any $\alpha, \beta\in (0, 1/L^2)$, $C_{\alpha, L}$ and $C_{\beta, L}$ essentially have the same self-similar subsets. Precisely, $E$ is a self-similar subset of $C_{\alpha,L}$ if and only if $\pi_{\beta}\circ \pi_\alpha^{-1}(E)$ is a self-similar subset of $C_{\beta,L}$, where $\pi_\alpha$ (similarly $\pi_\beta$) is the coding map from the symbolic space $\{0,1,\dots, L-1\}^\infty$ to $\mathcal C_{\alpha,L}$.
报告七摘要：
It is known that the Sierpinski measures are spectral ones. In this paper we study the spectral eigenmatri problems on Sierpinski measures $\mu$. That is (1) find all matric $A$, which satisfies that there exists a set $\Lambda$ such that both $\Lambda$ and $A\Lambda$ are spectra of $\mu$; And (2) given the model spectrum $\Lambda$, find all matrix $A$ such that $A\Lambda$ is a spectrum of $\mu$. Two criteria for $A$ to be a spectral eigenmatrix are given in this paper.
报告八摘要：
By introducing a ubiquity property for rectangles, we prove the mass transference principle from rectangles to rectangles.