Rigidity theorems in complex dynamics

讲座时间 Datetime: 2022年7月6日，星期三，15:00—16:30

地点 Venue:腾讯线上会议：321-103-462

主持人 Host：邵国宽

报告人 Speaker: 谢俊逸

单位 Affiliation:北京大学北京国际数学研究中心

报告摘要 Abstract:

Recently, with Zhuchao Ji, we answered several conjectures and questions about multiplier spectrum  of rational maps and giving new proofs of several  rigidity theorems in complex dynamics, by combining tools from complex and non-archimedean dynamics.

A remarkable theorem due to McMullen asserts that aside from the flexible Lattès family, the multiplier spectrum of periodic points determines the conjugacy class of rational maps up to finitely many choices. The proof relies on Thurston's rigidity theorem for post-critically finite maps, in where Teichmüller theory is an essential tool.  We will give a new proof of McMullen's theorem without using quasiconformal maps or Teichmüller theory.

We show that aside from the flexible Lattès family, the length spectrum of periodic points determines the conjugacy class of rational maps up to finitely many choices.  This generalize the aforementioned McMullen's theorem. We will also prove a rigidity theorem for marked length spectrum. Similar ideas  also yield a simple proof of a rigidity theorem due to Zdunik.

We show that a rational map is exceptional if and only if one of the following holds (i) the multipliers of periodic points are contained in the integer ring of an imaginary quadratic field;  (ii) all but finitely many periodic points have the same Lyapunov exponent.  This solves two conjectures of Milnor.