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

José M. Amigó; Ángel Giménez

doi:10.3934/dcdsb.2015.20.3415

Article Contents

2015, Volume 20, Issue 10: 3415-3434. Doi: 10.3934/dcdsb.2015.20.3415

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

José M. Amigó^1, and
Ángel Giménez^1,

1.
Universidad Miguel Hernández, Centro de Investigación Operativa, Avda. Universidad s/n, Elche (Alicante), 03202

Received: November 30, 2014

Revised: February 28, 2015

Published: November 2015

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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.

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Mathematics Subject Classification: Primary: 37B40, 37E05; Secondary: 37B10, 65Dxx.

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