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.