Encontros IST-IME 

05/09/2016, 15:40 — 16:20 — Anfiteatro Pa2, Pavilhão de Matemática
Fábio Tal, IME, Universidade de São Paulo

Zero entropy homeomorphisms of the sphere

We use a newly developed theory of forcing for surface homeomorphisms to obtain a Poincaré-Bendixson like result for orientation preserving homeomorphisms of the 2-sphere with zero topological entropy.

If $f$ is such a map and is not a pseudo-rotation, we show that for every $x$, there exists a power of $f$ such that the omega limit of $x$ must be either:

1. A cycle made of the union of unlinked fixed points and points heteroclinic to them.
2. A set rotating with irrational speed around a fixed point and possibly this fixed point.
3. An "infinitely renormalizable" set where the restriction of the dynamics is semi-conjugate to the odometer.

Joint work with P. Le Calvez.