Next Seminar
Survivor sets of Gauss map with a hole at $1$
Lingmin Liao 廖灵敏 (University of Wuhan 武汉大学)
2026-04-15 (Wed) 15:00 — Room 102, SCMS
Abstract: Given $\alpha \in [0,1]$, we investigate the set of numbers sharing identical representation of regular continued fractions and $\alpha$-continued fractions. We prove that modulo a countable set such a set is a survivor set of the Gauss map with a hole at $1$, i.e., the set of points $x$ such that all the iterations under Gauss map of $x$ is less than $\alpha$. Hence, these two sets have the same Haudorff dimension. We further show that with respect to $\alpha$, the function of such Hausdorff dimensions is increasing and locally constant almost everywhere. Moreover, we show that the function is not continuous at $0$, which is a new phenomenon in the study of open dynamical systems. This is a joint work with Cheng LIU.
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