Title 1: Dimensions of projected sets and measures on typical self-affine sets
Speaker: Pro.Dejun Feng(The Chinese University of Hong Kong)
Time: Mon, Dec.24, 2018 , AM:8:30-9:20
Location: Room 4131, Building No.4, Wushan Campus
Abstract：
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.
Title 2: On small bases which admit countably many expansions with multiple digits
Speaker: Pro.Bo Tan(Huazhong University of Science and Technology)
Time: Mon, Dec.24, 2018 , AM:9:20-10:10
Location: Room 4131, Building No.4, Wushan Campus
Abstract：
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 $$q^6=(m-1)q^5+(2m-1)q^4+(2m-1)q^3+2mq^2+mq+1.$$
Title 3: Dynamical dimension transference principle for dynamical Diophantine approximation
Speaker: Pro.Baowei Wang(Huazhong University of Science and Technology)
Time: Mon, Dec.24, 2018 , AM:10:20-11:10
Location: Room 4131, Building No.4, Wushan Campus
Abstract：
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
Title 4: On dichotomy law for beta-dynamical system in parameter space
Speaker: Pro.Jun Wu(Huazhong University of Science and Technology)
Time: Mon, Dec.24, 2018 , AM:11:10-12:00
Location: Room 4131, Building No.4, Wushan Campus
Abstract：
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
