Rotational entropy - a homotopy invariant for torus maps

To connect topological entropy with rotation theory, Botelho [2] introduced the notion of topological rotational entropy for annulus maps which are homotopic to the identity and later Geller and Misiurewicz [5] showed that it indeed vanished. In this paper, we generalize the notion of rotational entropy to any torus map, give its calculation formula and hence show that it is a homotopy invariant. We also introduce a notion of measure-theoretic rotational entropy and show that it is always less than or equal to the rotational entropy. (c) 2023 Elsevier Inc. All rights reserved.