Rigorous computation of topological entropy with respect to a finite partition

We present a method to compute rigorous upper bounds for the topological entropy h(T,A) of a continuous map T with respect to a fixed (coarse) partition of the phase space A. Long trajectories are not used; rather a single application of T to the phase space produces a topological Markov chain which contains all orbits of T, plus some additional spurious orbits. By considering the Markov chain as a directed graph, and labelling the arcs according to the fixed partition, one constructs a sofic shift with topological entropy greater than or equal to h(T,A). To exactly compute the entropy of the sofic shift, we produce a subshift of finite type with equal entropy via a standard technique; the exact entropy calculation for subshifts is then straightforward. We prove that the upper bounds converge monotonically to h(T,A) as the topological Markov chains become increasingly accurate. The entire procedure is completely automatic.