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 -