Topic Course on Dynamical Systems (No. 00103657)

Location: Teaching Building No. 3 (三教) Room 506.

Time: Wed. 10:10-12:00 (Odd-numbered weeks); Fri. 13:00-14:50 (All weeks)

Course description:

The modern theory of complex dynamics has rich intersections with analysis, geometry, topology, ergodic theory, geometric group theory, self-similar groups, graph theory, statistical mechanics, algebraic geometry, and even number theory. Many great mathematicians have contributed to this field, including Avila, Carleson, Milnor, McMullen, Smirnov, Sullivan, Thurston, Yoccoz, to just name a few Fields medal and Abel prize laureates who have made significant contributions.

In this course, we are going to focus on a class of fundamental objects in complex dynamics dating back to the works of W. P. Thurston, namely, post-critically finite branched covering maps on the topological 2-sphere. These maps, called Thurston maps nowadays, play a key role in Thurston’s characterization theorem of post-critically finite rational maps (known as the Fundamental Theorem of Complex Dynamics by some). The investigations on a class of Thurston maps with especially rich properties, called expanding Thurston maps, were initiated by the systematic study of M. Bonk and D. Meyer, as well as the related work of P. Haïssinsky and K. M. Pilgrim. We will take their monographs as our main references in this course.

A student in this course will be assigned a grade based on a survey report and a presentation on a topic in the intersection of complex dynamics with a direction listed above. The student can choose the direction that best suits his/her interests. A more ambitious student can consider a subsequent research project on the same topic.

This course is taught in English this semester, as part of a courses-taught-in-English program in PKU. The survey reports and presentations will be in English as well.

Prerequisites:

Real analysis, point set topology. Prior knowledge of functions of one complex variable is helpful, but not necessary.

References:

[1] Mario Bonk, Daniel Meyer, Expanding Thurston maps, Mathematical Surveys and Monographs, Vol. 225, Amer. Math. Soc., Providence, RI, 2017, xvi+478 pp.

[2] Peter Haïssinsky, Kevin M. Pilgrim, Coarse expanding conformal dynamics. Astérisque No. 325 (2009), viii+139 pp.