Professors: Andres Koropecki and Meysam Nassiri

Dates: September 02 - 13, 2013

The aim of this mini-course is to study the theory of prime ends and

its applications in two-dimensional dynamics, particularly in the

area-preserving setting.

Program:

1. Surfaces, topological ends.

2. Carathéodory's prime ends compactification of a bounded topological disk.

3. Extensions to ends of non-simply connected or unbounded sets.

4. Dynamics of prime ends, rotation numbers, Cartwright-Littlewood's

theorem [CL]. Examples [W].

5. Converse of Cartwright-Littlewood's theorem [KLN].

6. Generalizations to arbitrary surfaces and sets. Applications.

7. The C^r-generic area-preserving setting: irrationality of the prime

ends rotation number [M], density of stable/unstable manifolds of

periodic points in the sphere [FLC], and their generalizations.

8. Aperiodicity and topological properties of the boundary of

invariant open sets [KLN].

References:

[CL] M. L. Cartwright and J. E. Littlewood, Some fixed point

theorems, Ann. of Math. (2) 54 (1951), 1–37

[FLC] J. Franks and P. Le Calvez, Regions of instability for

non-twist maps, Ergodic Theory Dynam. Systems 23 (2003), no. 1,

111–141.

[KLN] A. Koropecki, P. Le Calvez, and M. Nassiri, Prime ends

rotation numbers and periodic points, eprint ArXiv:1206.4707 (2012)

[M] J. N. Mather, Invariant subsets for area preserving

homeomorphisms of surfaces, Mathematical analysis and applications,

Part B, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York,

1981, pp. 531–562.

[W] R. B. Walker, Periodicity and decomposability of basin

boundaries with irrational maps on prime ends, Trans. Amer. Math. Soc.

324 (1991), no. 1, 303–317.