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Transitivity, the existence of periodic points and positive topologicalentropy can be used to characterize complexity in dynamical systems. It isknown that for graphs that are not trees, for every $\varepsilon>0,$ thereexist (complicate) totally transitive maps (then with cofinite set of periods)such that the topological entropy is smaller than $\varepsilon$ (simplicity).First we will show by means of three examples that for any graph that is not atree the relatively simple maps (with small entropy) which are totallytransitive (and hence robustly complicate) can be constructed so that the setof periods is also relatively simple. To numerically measure the complexity ofthe set of periods we introduce a notion of a $\textit{boundary ofcofiniteness}$. Larger boundary of cofiniteness means simpler set of periods.With the help of the notion of boundary of cofiniteness we can state preciselywhat do we mean by extending the entropy simplicity result to the set ofperiods: $\textit{there exist relatively simple maps such that the boundary ofcofiniteness is arbitrarily large (simplicity) which are totally transitive(and hence robustly complicate)}$. Moreover, we will show that, the abovestatement holds for arbitrary continuous degree one circle maps.

Topological entropy, sets of periods and transitivity for graph maps