报告题目：Rotation Theory：Rotation Theory and Topological Dynamics

报告人 刘小川（IMPA）


报告时间 2017年5月3日10:00-11:00

报告地点  维格堂319

报告摘要  We introduce the theory of rotations in different contexts. While the initial purpose of the theory is to generalize classical Poincare theory on $S^1$, the development of this theory proves to be rich enough to become an independent research topic in topological dynamics. It surely helps to understand topological dynamics on surfaces of all kinds (different genus, with or without boundary). It also inspires many interesting theory to develop over the last decade. The first talk will include some very basic definitions and state some basic problems in this area. 

We show the important properties of a rotation set, which is closeness, connectedness and convexity. The last fact is the most non-trivial one. We will give a sketch of the proof. Finally, we introduce only briefly some tools that are very commonly used in this field, by mentioning some interesting works in recent years.


报告题目：Rotation Theory：The Case of Annulus

报告人 刘小川（IMPA）


报告时间 2017年5月3日11:00-12:00

报告地点  维格堂319

报告摘要  We start by reviewing classical rotation theory of Poincare for homeomorphisms on $S^1$. We stress the construction of Denjoy example, which is very important for constructing many examples later on, in general context.

In the first generalization to circle endomorphism, it provides a very simple example where we can get an interval as a rotation set. In the case of annulus, the difference of open and closed annulus should be stressed. The definitions for closed annulus are more or less the same with torus, but for open annulus, some more subtle problems exist.

We want to state the inspiring Boyland's conjecture, which was solved very recently via forcing theory. Here we state briefly one of the first attempts to solve this conjecture in. The tools developed turns out to be useful somewhere else. This attempt also shed light in some directions for understanding rotation theory on torus.


报告题目：Rotation Theory：Realization Problems

报告人 刘小川（IMPA）


报告时间 2017年5月4日09:00-10:00

报告地点  维格堂319

报告摘要  Still in the case of annulus, we will start by giving a detailed explanation of Franks' lemma on periodic free disk chains. The method has a connection with Brouwer theory for planar homeomorphisms. It also generalizes the Poincare-Birkhoff theorem. These ideas have been used repeatedly under very different circumstances over the decades. We mention the well-definedness of rotation number in infinite annulus. We mention another interesting and surprising modern application for rigidity of sphere (or annulus) diffeomorphisms, which preclude the property of mixing. 

Then we go on to prove realizing theorem in the case of torus. We start with Conley-Zenhder theorem, and then move to theorems of Franks.

The realization problem sometimes also involves Nielsen-Thurston classification of surface homeomorphisms. This is sort of a black box. We only briefly describe the situation and try to ask several questions which may excite future developments.



报告题目：Rotation Theory：The shape of Rotation Sets I.1:  Semi-Conjugacy

报告人 刘小川（IMPA）


报告时间 2017年5月4日10:00-11:00

报告地点  维格堂319

报告摘要  In order to recover a theory that should be similar to the circle case, a first try is to find semi-conjugacy to the corresponding rotation. Since we are in dimension two, we can also try to find semi-conjugacy to one-dimensional rotation, and then to reduce the problem to skew-products. An example shows this could not be done in general and a non-wandering condition should be added. 

In the minimal case, when the rotation direction is rational, the existence of a semi-conjugacy is a consequence of the famous Gottschalk-Hedlund theorem, under the condition of bounded deviation condition. 
This condition is one of the oldest and interesting conditions in this area. We will stress  this by some more examples.

One open problem which seems to be reachable is the uniqueness of minimal set in this situation. We formulate the problem in non-wandering case, and describe Avila's counter-example to the general case.


报告题目：Rotation Theory：The Shape of Rotation Sets I.2:  A Single Rational Vector

报告人 刘小川（IMPA）


报告时间 2017年5月4日11:00-12:00

报告地点  维格堂319

报告摘要  We start by stating a result of Le Calvez and Tal, which motivates our understanding for irrotational homeomorphisms.

We start by introducing the topological concept continuum. We give first examples of objects such that buckethandle and  the lake of Wada. These are indecomposable continuum. The lake of Wada is has the property where three disjoint open connected disks share the same boundary. Compare the Jordan curve theorem, which means the boundary of the lakes could not be a Jordan curve. But to understand this kind of compact set as a boundary of a topological disk, we refer to the last two lectures where we introduce prime ends.

To study homeomorphisms whose rotation set is a single rational rotation number, it reduces to the case where the homeomorphisms has a single rotation number $0/in /Bbb R^2$. $0$ rotation vector gives the impression that the dynamics does not move too far away, because it is a fact that each point does not move linearly to any directions. But, kind of surprisingly, there exists $C^/infty$ example for which almost every orbit approaches infinity in all directions. Of course they go to infinity with sub-linear speed. In particular there is no bounded deviation in any direction.

报告题目：Rotation Theory：The shape of Rotation sets II: Rotation Set with Non-empty Interior

报告人 刘小川（IMPA）


报告时间 2017年5月5日09:00-10:00

报告地点  维格堂319

报告摘要  Still in the case of annulus, we will start by giving a detailed explanation of Franks' lemma on periodic free disk chains (/cite{free_disk}). The method has a connection with Brouwer theory for planar homeomorphisms. It also generalizes the Poincare-Birkhoff theorem. These ideas have been used repeatedly under very different circumstances over the decades. We mention the well-definedness of rotation number in infinite annulus. We mention another interesting and surprising modern application for rigidity of sphere (or annulus) diffeomorphisms, which preclude the property of mixing. 

Then we go on to prove realizing theorem in the case of torus. We start with Conley-Zenhder theorem, and then move to theorems of Franks.

The realization problem sometimes also involves Nielsen-Thurston classification of surface homeomorphisms. This is sort of a black box. We only briefly describe the situation and try to ask several questions which may excite future developments.



报告题目：Rotation Theory：The shape of Rotation Sets III.1: Rotation Set with a Vertical Line Segment

报告人 刘小川（IMPA）


报告时间 2017年5月5日10:00-11:00

报告地点  维格堂319

报告摘要  When the rotation set is a non-trivial vertical line segment passing through the origin, then some lift to the universal covering has uniformly bounded horizontal deviation and the dynamics is also called annular. To use the condition of bounded or unbounded deviations, one can introduce Birkhoff stable set at infinity. The understanding of this set involves very technical and detailed arguments about the lifted homeomorphism on $/Bbb R^2$. 

Then, the focus is put to the famous Franks-Misiurewicz conjecture. The situation is very subtle, since three sub-cases of this conjecture has right now very different fate. We describe briefly two other cases.


报告题目：Rotation Theory：The shape of Rotation Sets III.2: Avila's Example with a Line Segment

报告人 刘小川（IMPA）


报告时间 2017年5月5日11:00-12:00

报告地点  维格堂319

报告摘要  We treat with care the third case of Franks-Misiurewicz conjecture. As mentioned in the previous lecture, Avila gave a counter-example at the end of 2013. It is the goal of this lecture to explain carefully Avila's example. The technique uses a variant of Anosov-Katok method, and the example obtained is in fact $C^/infty$. 

As a further variant of the method, we can also construct an interesting example, where for Lebesgue almost every point, the point-wise rotation number is not well-defined, i.e., the limit for the definition does not exist for almost every point. This is surprising if compared with the famous Oxtoby-Ulam Theorem in the 40s'. 


报告题目：Rotation Theory：The theory of Prime Ends I: Definitions with Examples

报告人 刘小川（IMPA）


报告时间 2017年5月11日10:00-11:00

报告地点  维格堂319

报告摘要  The purpose of this lecture is the explain the so called Caratheodory compactification of a simply connected domain, or, prime end compactification. 
We introduce the definitions of chains, cross cuts, equivalence of chains, and finally the definition of a prime end. Then, one can naturally define the prime end rotation number by looking at the dynamics.

In order to get good feeling of purpose of these definitions, we describe examples, in a more and more non-trivial order. 
The list of examples are as follows. We start with a simple example of unit circle attached with three hairs. Then we look at Matsumoto's examples. Finally, we consider Walker's examples, as well as the well-known example of pseudo-circles introduced by Bing.  For the final example, we will talk about Handel's treatment.


报告题目：Rotation Theory：The theory of Prime Ends II: some recent results

报告人 刘小川（IMPA）


报告时间 2017年5月11日11:00-12:00

报告地点  维格堂319

报告摘要  The introduction of prime end is a way to deal with complicated boundary of a simply connected domain. It is a kind of compactification by attaching ideal abstract elements which is homeomorphic to the unit circle. The theory of Poincare for circle homeomorphisms is again an important inspiration to understand the dynamics of the complicated topological disk via prime ends. However, it turns out to be quite easy to construct both the example with rational prime end rotation number with no periodic orbit on the original boundary as well as example with irrational prime end rotation number with periodic orbits on the boundary (even identity on the boundary).

The key observation for all such examples is the dissipativity. In the conservative case, some Poincare-like theory can be recovered and this is the main motivation of some recent works. This inquiry has a history, dating back to the Cartwright-Littlewood fixed point theorem, in the 50s', but more progress was made only recently precisely because of lack of new motivating interesting examples, which only appeared recently.