TY - JOUR
T1 - Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity
AU - Llibre, J.
AU - Mackay, R. S.
PY - 1991/1/1
Y1 - 1991/1/1
N2 - 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 ρ of the convex hull Δ of their rotation vectors, there is an orbit of rotation vector ρ, for each rational point p/q, p ℤ2, q ℕ, in the interior of Δ, there is a periodic orbit of rotation vector p / q, and for every compact connected subset C of Δ there is an orbit whose rotation set is C. Finally, we prove that f has ‘toroidal chaos’. © 1991, Cambridge University Press. All rights reserved.
AB - 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 ρ of the convex hull Δ of their rotation vectors, there is an orbit of rotation vector ρ, for each rational point p/q, p ℤ2, q ℕ, in the interior of Δ, there is a periodic orbit of rotation vector p / q, and for every compact connected subset C of Δ there is an orbit whose rotation set is C. Finally, we prove that f has ‘toroidal chaos’. © 1991, Cambridge University Press. All rights reserved.
U2 - https://doi.org/10.1017/S0143385700006040
DO - https://doi.org/10.1017/S0143385700006040
M3 - Article
VL - 11
SP - 115
EP - 128
JO - Ergodic Theory and Dynamical Systems
JF - Ergodic Theory and Dynamical Systems
SN - 0143-3857
ER -