Formulas for the topological entropy of multimodal maps based on min-max symbols
José M. Amigó - Universidad Miguel Hernández, Centro de Investigación Operativa, Avda. Universidad s/n, Elche (Alicante), 03202, Spain (email)
Abstract: In this paper, a new formula for the topological entropy of a multimodal map $f$ is derived, and some basic properties are studied. By a formula we mean an analytical expression leading to a numerical algorithm; by a multimodal map we mean a continuous interval self-map which is strictly monotonic in a finite number of subintervals. The main feature of this formula is that it involves the min-max symbols of $f$, which are closely related to its kneading symbols. This way we continue our pursuit of finding expressions for the topological entropy of continuous multimodal maps based on min-max symbols. As in previous cases, which will be also reviewed, the main geometrical ingredients of the new formula are the numbers of transversal crossings of the graph of $f$ and its iterates with the so-called "critical lines". The theoretical and practical underpinnings are worked out with the family of logistic parabolas and numerical simulations.
Keywords: Topological entropy, multimodal maps, symbolic dynamics, min-max symbols, algorithms.
Received: December 2014; Revised: March 2015; Available Online: September 2015.
2015 Impact Factor1.227