Date: Sat, Apr.13, 2019
Location: Room 3A01, Building No.37, Wushan Campus
Title1: H-Tolerance and Its Properties
Speaker: Pro.Zhiying Wen（Tsinghua University）
Time: AM.09:00-09:40
Abstract:
As a special external measure, H-tolerance has some special properties. The report describes the basic properties of H-tolerance, its relationship with other measures and some simple applications.
Title2: Estimates on the Dimension of Self-Similar Measures with Overlaps
Speaker: Pro.Dejun Feng（Chinese University of Hong Kong）
Time: AM.09:40-10:20
Abstract:
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.
Title3: Estimates on the Dimension of Self-Similar Measures with Overlaps
Speaker: Pro.Lifeng Xi（Ningbo University）
Time: AM.10:20-11:20
Abstract:
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.
Title4: Some Topology on Self-Affine Necklaces
Speaker: Pro.Shengyou Wen（Hubei University）
Time: AM.11:20-12:00
Abstract:
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.
Title5: Full Cylinders and Diophantine Analysis of Beta-Dynamical System in Parameter Space
Speaker: Pro.Jun Wu（Huazhong University of Science and Technology）
Time: PM.14:30-15:10
Abstract:
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.
Title6: Rigidity of Cantor Sets from the Point View of Self-Similar Subsets
Speaker: Pro.Hui Rao（Huazhong Normal University）
Time: PM.15:10-15:50
Abstract:
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 $\mathcal C_{\alpha,L}$.
Title7: The Spectral Eigenmatrix Problems of the Sierpinski Measures
Speaker: Pro.Xinggang He（Huazhong Normal University）
Time: PM.16:10-16:50
Abstract:
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.
Title8: Mass Transference Principle from Rectangles to Rectangles
Speaker: Pro.Baowei Wang（Huazhong University of Science and Technology）
Time: PM.16:50-17:30
Abstract:
By introducing a ubiquity property for rectangles, we prove the mass transference principle from rectangles to rectangles.
