A note on the periodic orbits and topological entropy of graph maps (Q2723490)

Let \(f\) be a self-map of a finite graph \(G\) (a compact connected Hausdorff space) so that any connected component of \(G\setminus V\) \((V\) is the set of vertices and finite) is homeomorphic to an open interval. Let \(A\subset G\) be finite and forward invariant under \(f\). Two such tripels \((G,A,f)\) and \((G',A',f')\) are called equivalent, if there is a conjugacy \(\Phi:G\to G'\) with \(\Phi(A)=A'\). Let \(h(f')\) denote topological entropy and define NEWLINE\[NEWLINEh\bigl( [G,A,f]\bigr): =\inf\bigl\{h(f'): (G,A,f)\sim (G',A',f')\bigr\}.NEWLINE\]NEWLINE The paper contains a proof of the statement that for any \(m\geq 1\) NEWLINE\[NEWLINEh(f)=\sup \biggl\{h \bigl([G,P,f]\bigl): P\text{ periodic orbit of length }>m\biggr\}.NEWLINE\]NEWLINE The first result of this type is due to \textit{Y. Takahashi} [Sci. Pap. Coll. Gen. Educ., Univ. Tokyo 80, 11-22 (1980; Zbl 0465.58023)].