学术报告报告主题：Matching for linear mod one transformations
报 告 人：Wolfgang Steiner 教授（法国巴黎西岱大学）
报告时间：2024年 10月15日（星期二）下午15:00-16:00
报告地点：34号楼602教室
邀 请 人：李兵 教授
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数学学院
2024年 10月11日
报告摘要：
For a linear mod one transformation (also called intermediate beta-transformation) $T_{\beta,\alpha}(x)= \beta x + \alpha - \lfloor \beta x + \alpha\rfloor$, matching holds if $T_{\beta,\alpha}^n(0) = T_{\beta,\alpha}^n(1^-)$ for some $n$. In this case, the absolutely continuous invariant measure has piecewise constant density. Bruin, Carminati and Kalle (2017) proved that matching occurs for almost all $\alpha$ when the base $\beta$ is a quadratic Pisot number or the Tribonacci number, and they conjecture that this holds for all Pisot numbers. Sun, Li and Ding (2023) have established relations with intermediate $\beta$-shifts of finite type and proved results on the fiber density. We discuss for which bases matching can occur and show that 0 is an accumulation point of matching parameters when $\beta$ is a Pisot number or a simple Parry number. We also discuss matching properties for alpha-continued fractions.
专家简介：
Wolfgang Steiner is Austrian and has obtained his PhD with Michael Drmota in Vienna. Since 2005, he has a CNRS research position at the Institut de Recherche en Informatique Fondamentale (IRIF) in Paris. He is mainly interested in various aspects of numeration systems (dynamical, combinatorial, number theoretical, geometrical, ...), in particular of beta-expansions and continued fractions. He is the main organiser of the (online) One World Numeration Seminar.