91直播

91直播：Karma Dajani  教授  (Utrecht University, The Netherlands)

时间：2025年05月13日  下午14:30—15:30; 15:40—16:40

          2025年05月14日  下午14:30—15:30; 15:40—16:40

          2025年05月15日  下午14:30—15:30; 15:40—16:40

地点：数统91直播 LD106

摘要：In this course we will study two dynamical systems associated with continued fraction expansions of numbers. In Chapter 1, we discuss the Gauss system whose iterates generate the continued fraction expansion of real numbers. We prove that the system is exact (hence ergodic and mixing). This allows us to use the ergodic theorem to reprove classic results such as Levi's Theorem on the growth of denominators of the rational approximations, the rationals obtained by truncating the infinite expansion at level $n$. We then move to the natural extension and show how one can extract the distribution of the approximation coefficients, quantities that measure the quality of approximation. In Chapter 2, we discuss the Ito-system, the dynamical system generating the Farey continued fractions (with digits 1 or 2).  We show it is ergodic, contains the (natural extension) of the Gauss map as a subsystem and generates rational approximations of a point containing the rationals associated with the regular continued fraction as well as the mediants.  We show how the dynamical action of inducing on a subregion allows one to pick any subsequence of approximating rationals, and to determine the distribution of the approximating coefficients associated with the subsequence.

具体内容见附件：

邀请人：数学研究中心

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