In 1984, Blokh proved [A. M. Blockh, On transitive mappings of one-dimensional branched manifolds, Differential-Difference Equations and Problems of Mathematical physics (Russian), Akad. Nauk Ukrain. SSR Inst. Mat., Kiev, 131, pp. 3-9, 1984] that any topologically transitive continuous map from a graph into itself which has periodic points has a dense set of periodic points and has positive topological entropy (in this proof a crucial role is played by the specification property, which implies these two statements). Also, he characterized the topologically transitive continuous graph maps without periodic points. Unfortunately, this clever paper is only available in Russian (except for a translation to English of the statements of the theorems without proofs - see [A. M. Blockh, The connection between entropy and transitivity for one-dimensional mappings, Uspekhi Mat, Nauk, 42(5(257)) (1987), pp. 209-210]). The present paper is an update on the basic properties of the topologically transitive graph maps with special emphasis on the density of the set of periodic points. In particular, we give full proofs of the aforementioned Blokh's results.
- Density of the set of periods
- Graph maps
- Topological entropy
- Topological transitivity