`a`
Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

Formulas for the topological entropy of multimodal maps based on min-max symbols

Pages: 3415 - 3434, Volume 20, Issue 10, December 2015      doi:10.3934/dcdsb.2015.20.3415

 
       Abstract        References        Full Text (511.5K)       Related Articles       

José M. Amigó - Universidad Miguel Hernández, Centro de Investigación Operativa, Avda. Universidad s/n, Elche (Alicante), 03202, Spain (email)
Ángel Giménez - 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.
Mathematics Subject Classification:  Primary: 37B40, 37E05; Secondary: 37B10, 65Dxx.

Received: December 2014;      Revised: March 2015;      Available Online: September 2015.

 References