报告人：曹春云博士 华中农业大学

报告题目: Dichotomy law for shrinking target problems in a nonautonomous dynamical system: Cantor series expansion

报告摘要：Let $Q=\{q_k\}_{k\geq 1}$ be a sequence of positive integers with $q_k\geq 2$ for every $k\geq 1$. Then each point $x\in [0,1]$ is attached with an infinite series expansion
 $$x=\frac{\varepsilon_1(x)}{q_1}+\frac{\varepsilon_2(x)}{q_1q_2}+\cdots+\frac{\varepsilon_n(x)}{q_1\cdots q_n}+\cdots,$$ which is called the Cantor series expansion of $x$. In this talk, we discuss the shrinking target problems for the system induced by the Cantor series expansion. More precisely, put $T_{Q}^{n}(x)=q_1\cdots q_nx-\lfloor q_1\cdots q_nx\rfloor$, the shrinking target problem in such a nonautonomous system can be formulated as considering the size of the set$$\mathbb{E}_y(\varphi):=\{x\in[0,1]:~ |T_{Q}^{n}(x)-y|<\varphi(n) \text{ i. o. }n\},$$
 where $y$ is a fixed point in $[0,1]$ and $\varphi: \mathbb{N}\to (0,1)$ is a positive function with $\varphi(n)\to 0$ as $n\to\infty$. It is proved that both the Lebesgue measure and the Hausdorff measure of $\mathbb{E}_{y}(\varphi)$ fulfill a dichotomy law according to the divergence or convergence of certain series.

报告时间：2017年7月10日(星期一)下午15:30-16:15

报告地点: 科技楼南楼602室