Transitivity and Dense Periodicity for Graph Maps

Abstract

Abstract 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]). Keywords: Topological transitivityGraph mapsDensity of the set of periodsTopological entropyPrimary 37E2537B20Secondary 37B40 Acknowledgements The first author has been partially supported by the DGES grant number PB96–1153, the second author has been partially supported by the grant number PGIDT 99 PXI20704B and the third one by the DGES grant number PB98-1574 and the FICYT grant number PB-EXP01-29. Notes †E-mail: [e-mail] ‡E-mail: [e-mail] ¶The terminology is not unified in the literature. Other authors, to denote this notion, use terms like regionally transitive, topologically ergodic, topologically indecomposable or topologically irreducible and nomadic, among others. §Usually (see for instance [21]) topological transitivity is considered the topological analogue of ergodicity. This terminology has been chosen following [21] by analogy with the notion of total ergodicity [3,27].