Prime ends and applications in two-dimensional dynamics

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.


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].


[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,
[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.