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. © 2001 Elsevier Science B.V. All rights reserved. MSC: 37B40; 94A17; 37M25

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