## Abstract

We present a numerical method to compute rigorous upper bounds for the topological entropy *h(T,A)* of a transitive 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.