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Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity. (English) Zbl 0699.58049
Ergodic Theory Dyn. Syst. (to appear).
We show that if a homeomorphism f of the torus, isotopic to the identity, has three or more periodic orbits with non-collinear rotation vectors, then it has positive topological entropy. Furthermore, for each point $$\rho$$ of the convex hull $$\Delta$$ of their rotation vectors, there is an orbit of rotation vector $$\rho$$, for each rational point p/q, $$p\in {\mathbb{Z}}^ 2$$, $$q\in {\mathbb{N}}$$, in the interior of $$\Delta$$, there is a periodic orbit of rotation vector p/q, and for every compact connected subset C of $$\Delta$$ there is an orbit whose rotation set is C. Finally, we prove that f has ‘toroidal chaos’.
Reviewer: J.Llibre

##### MSC:
 37A99 Ergodic theory 54C70 Entropy in general topology