报告人 刘小川教授 (巴西 IMPA 研究所)


地点：维格堂  319

2017.12.18  8：30-12：30


Lecture 1. We will introduce a general plan for this mini-course. There are three main aspects as follows. a)  rotation theory on torus. b) the paper “strictly toral dynamics” with a very recent new proof.
 c) The transverse foliation theory of P. Le Calvez. 

Lecture 2. In this hour, we will give some definitions and notations of the paper “strictly toral dynamics” by A. Koropecki and F. Tal.
 We try to at least give a good description of the statement of several main results of this paper, as well as a key technical “bounded disk lemma”
 for the proofs. 

Lecture 3. We continue to discuss the the main theorem of “strictly toral dynamics” paper, and give the proof based on our bounded disk lemma. 
We will also give some more motivations, with some examples.

Lecture 4. We give a very recent proof of the bounded disk lemma, put on Arxiv by Le calvez and Koropecki and Tal only last month. 

2017.12.19  9：00-12：00


Lecture 5. We introduce the notation of “rotation set” for homeomorphism of T^2. 
The big question of shapes of rotation sets motivates a lot of recent works. We will introduce several aspects of this theory 
and state a result providing some criterion for the possible realization of a compact and convex subset of R^2 as a rotation set. 

Lecture 6. We will start with a homeomorphism whose rotation set is a singleton. This is called a pseudo-rotation. There are some very natural questions in this case. It is the most interesting case comparing with rotation number
 of a circle homeomorphism. 

Lecture 7. We deal with a homeomorphism with a line segment as a rotation set. In this case, the most important problem is Franks-Misiurewicz conjecture. There are many developments in this direction, we try to describe some tools in this direction. 

2017.12.20  9：00-12：00

Lecture 8. We describe homeomorphism or diffeomorphism whose rotation set has non-empty interior. In this case, the dynamics is surely in the chaotic region. Therefore, this case is expected to be “generic”.

Lecture 9. Now we are in the third topic, the transverse foliation theory developed by Patrice Le calvez. 
The past ten years or so see great usefulness of this theory and it is still very useful in surface dynamics. 
In fact, the original proof of strictly total is by applying this foliation. 

Lecture 10. The proof of Le Calvez foliation is not very simple, it was published in IHES in 2006 and the paper is 98 pages. 
However, we will find a way to prove some partial results (rigorously), and give some applications, which show its power.  We plan to give a ten-hours mini-course. But just in case of extra time left, we might also give some idea and statements of the very recent preprint,
“Forcing Theory for transverse trajectories of surface homeomorphisms”, by P Le Calvez and F Tal. The starting point of this forcing theory is exactly transverse foliation.
 This paper was recently accepted Invent. Math.