Abstract

For a continuous map $f$ on the interval, to each periodic orbit $P$ of $f$ we can associate a number $h(P)$ which is the topological entropy of the connect-the-dots map corresponding to $P$ (or the linearization of $P$). In fact, $h(P)$ corresponds to the infimum of the entropies of all maps exhibiting orbits with the same combinatorics as P. Takahashi′s formula states that the entropy of $f$ is the supremum of the values $h(P)$ corresponding to all the periodic orbits $P$ of $f$ (furthermore, for each $n$, we can take the supremum only over the orbits whose size is greater than $n$). We show that an analogous relation between periodic behavior and topological entropy is satisfied for continuous maps on graphs.