学术报告报告题目：Dimensions of projected sets and measures on typical self-affine sets
报 告 人：丰德军 教授(香港中文大学)
报告时间：2018年12月24日（星期一）上午08:30-09:20
报告地点：4号楼131室
报告摘要：
In this talk, I will report some dimensional results on the projections of sets and measures on typical self-affine sets under the canonical coding maps. These are analogous to the classical dimensional results of Marstrand et al for orthogonal projections, but with different flavor. It is based on joint work with Chiu-Hong Lo.
报告题目：On small bases which admit countably many expansions with multiple digits
报 告 人：谭波 教授(华中科技大学)
报告时间：2018年12月24日（星期一）上午09:20-10:10
报告地点：4号楼131室
报告摘要：
Given a positive integer $M$, let $\mathcal{B}_{\aleph_0}(M)$ be the set of bases $q>1$ such that there exists a real number $x$ with exactly $\aleph_0$ different $q$-expansions over the alphabet $\{0,1,\ldots, M\}$. It is known that the smallest base in $\mathcal{B}_{\aleph_0}(M)$ is $\mathcal{G}(M)$, the generalized golden ratio. We investigate the next smallest element $q_{\aleph_0}(M)$ of $\mathcal{B}_{\aleph_0}(M)$, and show that if $M=2m$, $q_{\aleph_0}(M)$ is the appropriate root of $$q^3=mq^2+(m+1)q+1,$$ and if $M=2m-1$,$q_{\aleph_0}(M)$ is the appropriate root of $$q^6=(m-1)q^5+(2m-1)q^4+(2m-1)q^3+2mq^2+mq+1.$$
报告题目：Dynamical dimension transference principle for dynamical Diophantine approximation
报 告 人：王保伟 教授(华中科技大学)
报告时间：2018年12月24日（星期一）上午10:20-11:10
报告地点：4号楼131室
报告摘要：
Dynamical Diophantine approximation concerns the Diophantine properties of the orbit in a dynamical system. More precisely, let $(X,T)$ be a dynamical system with a metric $|\cdot|$. One concerns the size of the following limsup set defined via a dynamical system: $$W(\psi):=\Big\{x\in X: |T^nx-y|<\psi(n,x), \ {\text{i.o.}}, n\in \N\Big\}.$$ Following Hill \& Velani's pioneer work [Invent. Math. 95'], there have been many works done in concrete dynamical systems. We hope to find a general principle about the dimension of $W(\psi)$ in a general framework. By introducing a {\em dynamical ubiquity property}, it is shown that in an expanding exact topological dynamical system, when $\psi(n,x)=e^{-(f(x)+\cdots+f(T^{n-1}x))}$, both the dimension of $X$ and $W(\psi)$ are given by the solution to some pressure functions. While from the dimension of $X$ to that of $W(\psi)$, one needs only transfer the potential in the pressure equation. For this partial analogy with the mass transference principle in classic Diophantine approximation [Beresvenich \& Velani, Ann. of Math. 06'], we call the above phenomenon as a {\em dynamical dimension transference principle.}This is a joint with Guohua Zhang.
报告题目：On dichotomy law for beta-dynamical system in parameter space
报 告 人：吴军 教授(华中科技大学)
报告时间：2018年12月24日（星期一）上午11:10-12:00
报告地点：4号楼131室
报告摘要：
Let $\varphi\colon \mathbb{N}\rightarrow(0,1]$ be a positive function, we prove that the set \begin{equation*}E(0, \varphi)=\{\beta>1\colon |T^{n}_{\beta}1-0|<\varphi(n) \textrm{ for infinitely many } n\in\mathbb{N}\}\end{equation*} 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. As an application, we determine the exact Lebesgue measure of the set\begin{equation*}\mathfrak{E}(0, \{l_{n}\})=\{\beta>1\colon |T^{n}_{\beta}1-0|<\beta^{-l_{n}} \text{ for infinitely many } n\in\mathbb{N}\},\end{equation*} where $\{l_{n}\}_{n\geq 1}$ is a sequence of non-negative numbers.
邀 请 人: 李兵 教授
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数学学院
2018年12月21日