×

zbMATH — the first resource for mathematics

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