Computing Topological Entropy in a Space of Quartic Polynomials

This paper adds a computational approach to a previous theoretical result illustrating how the complexity of a simple dynamical system evolves under deformations. The algorithm targets topological entropy in the 2-dimensional family P ^Q of compositions of two logistic maps. Estimation of the topological entropy is made possible by the correspondence between P ^Q and a subfamily of sawtooth maps P ^T , and is based on the well-known fact that the kneading-data of a map determines its entropy. A complex search for kneading-data in P ^T turns out to be computationally fast and reliable, delivering good entropy estimates. Finally, the algorithm is used to produce a picture of the entropy level-sets in P ^Q , as illustration to theoretical results such as Hu (Ph.D. thesis, CUNY, 1995) and Radulescu (Discrete Cont. Dyn. Syst. 19(1):139–175, 2007).