The aim of this minicourse is to give an overview of results and techniques in two-dimensional dynamics from the topological viewpoint. Emphasis will be given to rotational invariants (rotation vectors and numbers) and their dynamical consequences.

The plan is to introduce the general results from Brouwer theory as well as Brouwer-Le Calvez foliations and their applications, and to make a survey of the results about rotation sets on the torus and annulus and their dynamical consequences.

We will also talk about prime ends rotation numbers, and their use in the study of the dynamics on the boundary of invariant sets. If time allows, we will discuss rotation numbers and intervals on more general invariant continua.

S. Addas-Zanata, Area-preserving diffeomorphisms of the torus whose rotation sets have non-empty interior, Erg. Th. & Dyn. Sys, 2015

M. L. Cartwright and J. E. Littlewood, Some fixed point theorems, Ann. of Math. (2) 54 1--37 (1951)

P. A. Guihéneuf, A. Koropecki, Stability of the rotation set of area-preserving toral homeomorphisms, Nonlinearity, to appear

J. Franks, Realizing rotation vectors for torus homeomorphisms, Trans. Amer. Math. Soc. 311 107--115 (1989)

A. Koropecki, Realizing rotation numbers on annular continua, Mathematische Zetschrift (2016)

A. Koropecki, P. Le Calvez, M. Nassiri, Prime ends rotation numbers and periodic points, Duke Math. J. 164 (2015), no. 3, pp. 403-472

A. Koropecki, A. Passeggi, A Poincaré-Bendixson theorem for translation lines and applications to prime ends, Preprint arXiv:1701.04644

A. Koropecki, F. A. Tal, Strictly toral dynamics, Inventiones Mathematicae 196 (2014), no. 2, pp. 339-381

A. Koropecki, F. A. Tal, Fully essential dynamics for area-preserving surface diffeomorphisms, Erg. Th. & Dyn. Sys, to appear

P. Le Calvez, Une version feuilletée équivariante du théorème de translation de Brouwer, Publ. Math. Inst. Hautes Études Sci. 1--98 (2005)

P. Le Calvez, F. A. Tal, Forcing theory for transverse trajectories of surface homeomorphisms, preprint arXiv:1503.09127

J. Mather, Topological proofs of some purely topological consequences of Carathéodory's theory of prime ends, Selected studies: physics-astrophysics, mathematics, history of science, North-Holland, 1982, pp. 225-255

J. Mather, Invariant subsets for area preserving homeomorphisms of surfaces, Mathematical analysis and applications, part B, Adv. in Math. Suppl. Stud. 7, Academic Press, 1981, 531-562.

M. Misiurewicz, K. Ziemian, Rotation sets for maps of tori, J. London Math. Soc. (2) 40 490--506 (1989)