Algorithms for Rigorous Entropy Bounds and Symbolic Dynamics

The aim of this paper is to introduce a method for computing rigorous lower bounds for topological entropy. The topological entropy of a dynamical system measures the number of trajectories that separate in finite time and quantifies the complexity of the system. Our method relies on extending existing computational Conley index techniques for constructing semiconjugate symbolic dynamical systems. Besides offering a description of the dynamics, the constructed symbol system allows for the computation of a lower bound for the topological entropy of the original system. Our overall goal is to construct symbolic dynamics that yield a high lower bound for entropy. The method described in this paper is algorithmic and, although it is computational, yields mathematically rigorous results. For illustration, we apply the method to the Hénon map, where we compute a rigorous lower bound of 0.4320 for topological entropy.