A closed formula for the topological entropy of multimodal maps based on min-max symbols

Topological entropy is a measure of complex dynamics. In this regard, multimodal maps play an important role when it comes to study low-dimensional chaotic dynamics or explain some features of higher dimensional complex dynamics with conceptually simple models. In the first part of this paper an analytical formula for the topological entropy of twice differentiable multimodal maps is derived, and some basic properties are studied. This expression involves the so-called min-max symbols, which are closely related to the kneading symbols. Furthermore, its proof leads to a numerical algorithm that simplifies a previous one also based on min-max symbols. In the second part of the paper this new algorithm is used to compute the topological entropy of different modal maps. Moreover, it compares favorably to the previous algorithm when computing the topological entropy of the bi- and tri-modal maps considered in the numerical simulations.