Polynomial entropy and rotation sets

We investigate the polynomial entropy of continuous maps on the d -dimensional torus Tᵈ T d ( d ≥ 1 d ≥ 1 ) that are homotopic to the identity. We establish that if the associated rotation set of such a map contains more than a single point, then its polynomial entropy is at least 1. As a corollary, we show that for any orientation-preserving homeomorphism f on the 2-dimensional torus with polynomial entropy strictly less than 1, there exists some q ∈ N q ∈ N such that the iterate fq f q is a pseudo-rotation; that is, fq f q is homotopic to the identity and its rotation set is a singleton.