Computing the topological entropy of unimodal maps

TL;DR

This paper presents a recursive algorithm to compute the lap number of unimodal maps, enabling precise estimation of their topological entropy and growth number through sign analysis of critical and boundary point itineraries.

Contribution

The authors introduce a novel recursive algorithm based on sign analysis for calculating the lap number of unimodal maps, advancing the understanding of their topological complexity.

Findings

The algorithm accurately computes the lap number of unimodal maps.

It effectively estimates the growth number and topological entropy.

Application to bifurcation maps demonstrates its practical utility.

Abstract

We derive an algorithm to determine recursively the lap number (minimal number of monotone pieces) of the iterates of unimodal maps of an interval with free end-points. The algorithm is obtained by the sign analysis of the itineraries of the critical point and of the boundary points of the interval map. We apply this algorithm to the estimation of the growth number and the topological entropy of maps with direct and reverse bifurcations.