Rotational Theory in Low Dimensions

1. Rotation theory on the circle

1.1 Circle homeomorphisms

– Rotation number for circle homeomorphisms.
– Existence of periodic orbits and uniqueness of minimal sets.
– Denjoy’s theorem and counter-examples.
– Smooth linearization of circle diffeomorphisms with Diophantine rotation number. (Arnold-Herman-Yoccoz theorem).

1.2 Degree 1 circle endomorphisms

– Pointwise rotation number and rotation set.
– Realizing rational numbers by periodic orbits.
– Endomorphisms with positive topological entropy.

2. Rotation theory on surfaces

2.1 Homeomorphisms homotopic to the identity

– Diverse rotation sets: pointwise, associated to ergodic measures and the Misiurewicz-Ziemian ones.
– Realizing rational vectors by periodic orbits.

– The rotation sets of flows on the torus.

– On the dynamics of homeomorphisms whose rotation sets have non-empty interior.
– On the dynamics of homeomorphisms whose rotation sets are singleton.
– On the dynamics of homeomorphisms whose rotation sets are line segments. The Franks-Misiurewicz conjecture.

2.2 Homeomorphisms of the torus homotopic to a Dehn twist

– Vertical rotation numbers and vertical rotation set.
– Realizing rational rotation numbers by periodic points.
– On the dynamics of periodic-point-free homeomorphisms in a Dehn twist homotopy class.

2.3 Some notions of rotation theory on other surfaces.

Referencias:
Edson de Faria, Pablo Guarino. Dynamics of Circle Mappings, Publicação do 33º Colóquio Brasileiro de Matemática, IMPA, (2023).
Michael Herman. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publications Mathématiques de l’IHÉS. Volume 49 (1979), 5-233.
Michał Misiurewicz, Krystyna Ziemian. Rotation Sets for Maps of Tori. Volumes 2-40, (1989), 490-506.
John Franks and Michał Misiurewicz. Rotation set of toral flows. Proc. Amer. Math. Soc. 109 (1990), 243-249.
Philip Boyland, Topological methods in surface dynamics. Volume 58, (1994), 223-298.
Jaume Llibre, Robert Mackay, Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity. Ergod. Th. & Dynam. Sys. (1991), vol 11, 115-128.
Pablo Dávalos, On torus homeomorphisms whose rotation set is an interval. Math. Z. (2013), 275, 1005-1045.
Alejandro Kocsard, On the dynamics of minimal homeomorphisms of T^2 which are not pseudo-rotations. Annales scientifiques de l’École normale supérieure (2021), 54, 991-103
Pierre-Antoine Guihéneuf, Théorie de forçage des homéomorphismes de surfaces [d’après Le Calvez et Tal], Séminaire Bourbaki 72 année, 2019-2020, 1171.