Topological entropy of one-dimensional maps: Approximations and bounds (original) (raw)

Computing the topological entropy of general one-dimensional maps

Trans. Am. Math. Soc, 1991

A matrix-theoretic method for computing the topological entropy of continuous, piecewise monotonic maps of the interval is presented. The method results in a constructive procedure which is easily implemented on the computer. Examples for families of unimodal, nonunimodal and discontinuous maps are presented.

Topological Entropy in a Parameter Range of the Standard Map

Progress of Theoretical Physics, 2009

We combine the trellis method and the braid method, and by estimating the lower bounds of the topological entropy of the standard map for a certain parameter range, we follow the change of the topological entropy. The trellis in a tangency situation of the stable and unstable manifolds is constructed. Applying the trellis method to this trellis, the lower bound of topological entropy is calculated. There exist systems in which the trellis method can not be applicable. For such systems, we look for non-Birkhoff periodic orbits existent in the trellises, form braids from those periodic orbits, and estimate the topological entropy from their braid types. We perform these tasks for a sequence of trellises and numerically visualize the change in the topological entropy. In addition, we take particular sequence of connecting orbits to obtain homoclinic or heteroclinic orbits. As a natural extension, we assign topological entropy to these homoclinic and heteroclinic orbits.

On the estimation of topological entropy

Journal of Statistical Physics, 1993

We study a method for estimating the topological entropy of a smooth dynamical system. Our method is based on estimating the logarithmic growth rates of suitably chosen curves in the system. We present two algorithms for this purpose and we analyze each according to its strengths and pitfalls. We also contrast these with a method based on the definition of topological entropy, using (n, e)spanning sets.