On torus homeomorphisms of which rotation sets have no interior points (Q675760)

The paper deals with a homeomorphism of a 2-dimensional torus which is isotopic to the identity. The rotation set \(\rho(f)\) of the homeomorphism \(f\) is defined through a lift of \(f\) to the universal cover. The rotation set characterizes an asymptotic behavior of the lift. \textit{J. Franks} [Trans. Am. Math. Soc. 311, No. 1, 107-115 (1989; Zbl 0664.58028)] showed that rational internal points of the rotation set are realized by periodic points. If \(\text{Int}\rho (f)=\emptyset\) and \(f\) preserves a Lebesgue measure, the same property holds. The main result of the paper is the following theorem. If the rotation set is a closed segment of irrational slope, and the chain recurrent set of \(f\) is not chain transitive, then the rotation set includes a rational point which is realized by a periodic point.